Device, system, and method for hemispheric breast imaging

ABSTRACT

A device, system, and method for volumetric ultrasound imaging is described. The device and system include an array of transducer elements grouped in triangular planar facets and substantially configured in the shape of a hemisphere to form a cup-shaped volumetric imaging region within the cavity of the hemisphere, A plurality of data-acquisition assemblies are connected to the transducers, which are configured to collect ultrasound signals received from the transducers and transmit image data to a network of processors that are configured to construct a volumetric image of an object within the imaging region based on the image data received from the data-acquisition assemblies.

RELATED APPLICATIONS

This application is a continuation of co-pending U.S. patent applicationSer. No. 15/501,792, filed Feb. 3, 2017, which is a national stageapplication of PCT Pat. Appl. No. PCT/US2015/043628, filed Aug. 4, 2015,which claims priority to U.S. Prov. Pat. Appl. No. 62/033,313, filedAug. 5, 2014. The entire contents of the aforementioned patentapplications are hereby incorporated by reference.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Grant Nos.EB009692 and EB010069 awarded by the National Institutes of Health(NIH). The government has certain rights in the invention.

BACKGROUND

Breast cancer is a significant health problem worldwide. In the UnitedStates alone, more than 230,000 new cases of invasive breast cancer areestimated to be diagnosed each year and about 40,000 women are expecteddie of the disease this year. Globally, when excluding non-melanomacancers of the skin, breast cancer is the most common cancer in women.

An important clinical goal is to detect breast masses when they are assmall as possible, preferably less than several millimeters in diameter.Reports indicate that women who have invasive breast cancer detectedwhen the size is less than 15 mm have a 15-year survival rate of 89-93%(95% confidence interval). Imaging is the primary way that cancer in thebreast can be detected when the cancer is small. In addition, imagingcan also be used for staging and monitoring response to the treatment ofa patient with breast cancer.

The breast can be imaged using a number of methods, includingconventional x-ray mammography, x-ray tomosynthesis, and magneticresonance imaging (MRI). However, current implementations of thesemethods often suffer from low resolution, poor contrast, or other issuesthat reduce the effectiveness of these techniques in detecting oridentifying breast disease.

For example, X-ray mammography is generally considered to be the mostcost-effective tool for the early detection of breast cancer. However,the specificity and positive predictive value of mammography is limited,due to the potential overlap in the appearances of benign and malignantlesions, and to poor contrast in patients with dense breast tissue.

Ultrasound is not typically used for the diagnosis of breast diseasebecause the process of obtaining the images is highly operatordependent. Further, ultrasound resolution is generally not adequate,particularly in the direction orthogonal to the imaging plane, i.e., theslice thickness dimension, and speckle can make images hard to interpretor can obscure calcifications. Current ultrasound techniques also oftenpoorly describe lesion margins that are known to be an important featurefor the diagnosis of cancer.

Accordingly, there is a continuing need in the art for imagingtechniques that can be used to image breasts accurately for the purposesof detecting or identifying breast disease.

SUMMARY

In one embodiment, a device for volumetric ultrasound imaging isclaimed. The device includes an array of ultrasound transducer elementssubstantially configured in the shape of a hemisphere to form acup-shaped volumetric imaging region within the cavity of thehemisphere. In another embodiment, the array of transducers includes 40triangular planar facets. In another embodiment, 10 of the facets areequilateral triangles and 30 of the facets are isosceles triangles. Inanother embodiment, each triangular transducers includes 256piezoelectric elements. In another embodiment, the piezoelectricelements are arranged pseudorandomly on each facet. In anotherembodiment, at least one of the transducers further includes a diverginglens. In another embodiment, at least one of the transducers furtherincludes two matching layers. In another embodiment, the hemispherearray of transducers is positioned within the surface of a patienttable, such that the opening of the cup-shaped volumetric imaging regionis substantially flush with the patient table surface. In anotherembodiment, the device further includes a cup-shaped container sized tofit substantially within the imaging region cavity of the hemisphere. Inanother embodiment, the cup-shaped container is disposable.

In another embodiment, a system for volumetric ultrasound imaging isclaimed. The system includes an array of planar faceted ultrasoundtransducers substantially configured in the shape of a hemisphere toform a cup-shaped volumetric imaging region within the cavity of thehemisphere, a plurality of data-acquisition assemblies connected to thetransducers, and a network of processors connected to thedata-acquisition assemblies. The ultrasound transducers are configuredto generate and receive ultrasound signals within the imaging region,the data-acquisition assemblies are configured to collect ultrasoundsignals received from the transducers and transmit measured data to thenetwork of processors, and the network of processors is configured toconstruct a volumetric image of an object within the imaging regionbased on the image data received from the data-acquisition assemblies.In another embodiment, the number of data-acquisition assemblies isequal to the number of transducer elements, and that eachdata-acquisition assembly is dedicated to an individual transducer. Inanother embodiment, the array of transducers comprises 40 triangularplanar faceted transducer subarrays. In another embodiment, 10 of thefacets are equilateral triangles and 30 of the facets are isoscelestriangles. In another embodiment, each transducer comprises 256piezoelectric elements. In another embodiment, each data-acquisitionassembly comprises at least 256 send/receive channels. In anotherembodiment, the network of processors comprises at least 20 nodes. Inanother embodiment, each node comprises at least one graphicalprocessing unit (GPU). In another embodiment, each node is configured toprocess data received from at least two data-acquisition assemblies inparallel.

In another embodiment, a method for reconstructing a volumetricultrasound image is claimed. The method includes the steps of generatingultrasound signals from an array of transducer elements substantiallyarranged in the shape of a hemisphere, such that the generated signalsare incident on a scattering object positioned within the cavity formedby the hemisphere of transducer elements; measuring a plurality ofscattered ultrasound signals with the ultrasound transducer elements;pre-processing the measured signals with a plurality of data-acquisitionassemblies; and reconstructing a volumetric image of the scatteringobject with a network of processors based on the pre-processed signals.In one embodiment, the image of the scattering object is reconstructedvia an inverse-scattering model.

In one embodiment, the image of the scattering object is reconstructedby relating the measured scattered ultrasound signals to backgroundmedium variation. In such an embodiment, an estimate image of thescattering object is reconstructed from a subset of the measuredsignals; one or more refined images of the scattering object arereconstructed from the estimate image of the scattering object andpre-processed signals; and the volumetric image of the scattering objectis reconstructed from the one or more refined images and pre-processedsignals. In one embodiment, the background medium variation issubstantially linear in relation to the measured signals.

In one embodiment, at least a portion of the scattering object data isinterpolated. In one such embodiment, the interpolation comprises aprefiltering step. In one embodiment, scattering measured in thehemisphere transducer array is used to estimate scattering object detailthat would come from scattering in the antipodal hemisphere. In oneembodiment, the measured signals are divided into overlappingsubvolumes. In one embodiment, the subvolumes are tetrahedral. In oneembodiment, the background medium variation of each subvolume iscalculated independently. In one embodiment, the calculation of thebackground medium variation of each subvolume is processed substantiallysimultaneously. In one embodiment, an array of background mediumvariation values corresponding to the overlapping subvolumes is rotatedduring reconstruction. In one embodiment, the background mediumvariation array is rotated to align an axis of the background mediumvariation array with an axis of a portion of the transducer elements. Inone embodiment, the rotation of the background medium variation array isfactored into one or more skew operations. In one embodiment, virtualscattering measurements from the background medium variations arecomputed at the vertices of a regular grid that span the surface of eachfacet. In one embodiment, the virtual scattering measurementcomputations further comprise an inverse two-dimensional Fast Fouriertransform of spatial frequency integrals. In one embodiment, the virtualscattering measurements are used to interpolate actual scatteringmeasurements at the locations of the facet elements. In one embodiment,the interpolations further comprise a prefiltering step.

In one embodiment, an estimate image of the scattering object isreconstructed from a subset of the measured signals; one or more refinedimages of the scattering object are reconstructed from the estimateimage and pre-processed signals corresponding to the overlappingsubvolumes of measured signals; and a volumetric image of the scatteringobject is reconstructed from the one or more refined images andpre-processed signals corresponding to the overlapping subvolumes ofmeasured signals.

In one embodiment, a time gate is applied to the measured signals priorto reconstruction to isolate the measured signals from differentsubvolumes.

BRIEF DESCRIPTION OF THE DRAWINGS

The following detailed description of embodiments will be betterunderstood when read in conjunction with the appended drawings. Itshould be understood, however, that the embodiments are not limited tothe precise arrangements and instrumentalities shown in the drawings.

FIG. 1 is a schematic of an exemplary embodiment of a hemispheric breastimaging system (HABIS) integrated with a patient examination table.

FIG. 2 is a schematic of another exemplary embodiment of a HABISintegrated with a patient examination table, wherein the examinationtable surface and a portion of the sides of the table have been removed.

FIG. 3 is a schematic of the front-end electronics of an exemplaryembodiment of a HABIS, showing a hemispheric cup and ultrasoundtransducer assemblies.

FIG. 4 , comprising FIGS. 4A through 4C, is a set of images of anexemplary embodiment of a transducer element subassembly, which can forma part of a hemispheric ultrasound transducer array. FIG. 4A is a photoimage of the inner surface of the transducer element subassembly, FIG.4B is a cross-sectional X-ray image of a portion of the subassembly, andFIG. 4C is a photo image of the outer surface of the subassembly.

FIG. 5 is a schematic of the front-end electronics of an exemplaryembodiment of a HABIS, showing a portion of the data-acquisitionsubassemblies.

FIG. 6 is a schematic of the front-end electronics of an exemplaryembodiment of a HABIS, showing a plurality of data-acquisitionsubassemblies connected to the ultrasound transducer assemblies.

FIG. 7 is a schematic of the front-end electronics of an exemplaryembodiment of a HABIS, showing how an exemplary data-acquisitionsubassembly is connected to an ultrasound transducer assembly.

FIG. 8 is a schematic of the AC wiring and power supply distribution ofan exemplary embodiment of a HABIS.

FIG. 9 is a schematic of the HABIS patient examination table of FIG. 1positioned in an exemplary examination room.

FIG. 10 is a schematic of a hemispheric volume (FIG. 10A) and thetetrahedral region that is associated with a subtriangle of thehemispheric array (FIG. 10B). The portion of the hemispheric volumeimaged by the data from the triangular block of receive elements in thetriangle is the tetrahedron formed by the vertices of the solid-linetriangle and the center of the hemisphere.

FIG. 11A is a schematic of an exemplary embodiment of a high-performancecomputer network communicatively connected to the hemispheric transducerarray of a HABIS.

FIG. 11B is an enlarged view of a portion of the high-performancecomputer network of FIG. 11A.

FIG. 12 is a diagram of a data processing flow through an exemplaryHABIS and associated high-performance computer network.

FIG. 13 is a diagram of a tetrahedral region associated with asubtriangle of the hemispheric array. Image values are computed atvertices of a set of cubic subvolumes with centers that intersects thetetrahedral subvolume.

FIG. 14 , comprising FIGS. 14 and 14B, is a diagram of thereconstruction of a 10-point resolution target (14A) and a correspondingb-scan of the target (14B). The surfaces shown are 50% of the maximumamplitude. The two upper-most points are 200 m apart, the two left-mostpoints are 300 m apart, and the two right-most points are 400 m apart.In each panel, an edge of the cubic volume has a length of 2.9 mm andthe tic increment on the axes is 0.5 mm.

FIG. 15 is a set of images showing sections of a representativesubvolume in a model of the breast. Columns left to right: Sections inthe original subvolume, corresponding sections in the inverse-scatteringreconstruction of the subvolume, and corresponding sections in theaberration-corrected b-scan of the subvolume. Rows top to bottom:Orthogonal relation between the subvolume sections, x-y plane at z=0,x-z plane at y=0, and y-z plane at x=0. Scattering throughout thehemisphere was used to estimate scattering obtained from a virtual arrayin the opposite hemisphere and then both sets of data were used in theinverse scattering reconstruction. The b-scan used reception from a64-element subtriangle and used transmission from that 64-elementsubtriangle and three surrounding 64-element subtriangles to allow quickformation of a volumetric image. In each panel, an edge of the sectionhas a length of 6.4 mm, and the outer region of the section is weightedby a window used to blend adjacent reconstructed subvolumes. The x, y,and z axes are denoted accordingly. The z axis is the polar axis of thehemisphere, the approximate direction of the b-scan lines which mainlyhighlight discontinuities perpendicular to them. The original subvolumeand the reconstructed subvolume are shown on a linear scale and theb-scan is shown on a log scale.

FIG. 16 illustrates coordinate transformation produced by skewing thehorizontal generating vector.

FIG. 17 illustrates three skew transformations that yield a rotatedcoordinate system.

FIG. 18 illustrates geometry used to determine skews that result in a θrotation.

FIG. 19 illustrates geometry of array size requirements for in-placerotation.

FIG. 20 illustrates planar rotations R_(θ) ² R_(ϕ) ³ satisfy R_(θ) ²R_(ϕ) ³u₃=sin θ cos ϕu₁+sin θ sin ϕu₂+cos ϕu₃.

FIG. 21 illustrates distortion curves of interpolations at fractionaloffsets of 0.1, 0.2, 0.3, and 0.4 using coefficients giving minimumleast-square error in the frequency band [−π, +π]. The interpolationfilter has coefficients.

FIG. 22 illustrates distortion curves of interpolations at fractionaloffsets of 0.1, 0.2, 0.3 and 0.4 using coefficients giving minimumleast-square error in the frequency band [−0.75π, +0.75π] The fourpanels show the distortion for interpolation filters with 6 (upperleft), 8 (upper right), 10 (lower left) and 12 (lower right)coefficients.

FIG. 23 illustrates rectangular boundaries of forward solutions fromtransmit elements in a facet that contain the upper hemisphere (dottedrectangles) and the additional boundaries associated with thedirectivity patterns of all the elements in the facet (trapezoidalregions).

FIG. 24 illustrates frequency-dependent distortion from interpolationsat different fractional offsets that result from four-pointinterpolations using weights that are determined by fitting thefrequency response of the interpolation to the time-shift responsee^(−jωε)

FIG. 25 illustrates frequency-dependent distortion from interpolationsat different fractional offsets that result from four-pointinterpolations using weights that are determined by fitting thefrequency response of the interpolation to the time-shift responseW_(α)(ω)e^(−jωε) and 1/W_(α)(ω) prefiltering.

FIG. 26 illustrates frequency dependent distortion from interpolationsat different fractional offsets that result from four-pointinterpolations using weights that are determined by fitting thefrequency response of the interpolation to the time-shift responseW_(α)(ω)e^(−jωε) over the frequency range [−0.8π, +0.8π] and 1/W_(α)(ω)prefiltering.

DETAILED DESCRIPTION

It is to be understood that the figures and descriptions have beensimplified to illustrate elements that are relevant for clearunderstanding, while eliminating, for the purpose of clarity, many otherelements found in the field of ultrasound imaging systems. Those ofordinary skill in the art may recognize that other elements and/or stepsare desirable and/or required in implementing the systems and methodsdescribed herein. However, because such elements and steps are wellknown in the art, and because they do not facilitate a betterunderstanding, a discussion of such elements and steps is not providedherein. The disclosure herein is directed to all such variations andmodifications to such elements and methods known to those skilled in theart.

Definitions

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art. Any methods and materials similar or equivalent to thosedescribed herein can be used in the practice for testing of the systemsand methods described herein. In describing and claiming the systems andmethods, the following terminology will be used.

It is also to be understood that the terminology used herein is for thepurpose of describing particular embodiments only, and is not intendedto be limiting.

The articles “a” and “an” are used herein to refer to one or to morethan one (i.e., to at least one) of the grammatical object of thearticle. By way of example, “an element” means one element or more thanone element.

“About” as used herein when referring to a measurable value such as anamount, a temporal duration, and the like, is meant to encompassvariations of 20%, ±10%, ±5%, ±1%, or +0.1% from the specified value, assuch variations are appropriate.

The terms “HABIS,” “system,” and the like are used interchangeablyherein and refer to a system comprising a hemispheric array ofultrasound transducers and a computer network suitable forhigh-performance parallel processing of data collected from thehemispheric array. As described herein, such a system may includesoftware and associated algorithms to reconstruct volumetric images of apatient's breast. It is also contemplated herein that such a system canbe configured to reconstruct volumetric images of other parts of apatient's anatomy, or any other scattering object.

The terms “patient,” “subject,” “individual,” and the like are usedinterchangeably herein, and refer to any animal amenable to the systems,devices, and methods described herein. Preferably, the patient, subjector individual is a mammal, and more preferably, a human.

Ranges: throughout this disclosure, various aspects can be presented ina range format. It should be understood that the description in rangeformat is merely for convenience and brevity and should not be construedas an inflexible limitation on the scope. Accordingly, the descriptionof a range should be considered to have specifically disclosed all thepossible subranges as well as individual numerical values within thatrange. For example, description of a range such as from 1 to 6 should beconsidered to have specifically disclosed subranges such as from 1 to 3,from 1 to 4, from 1 to 5, from 2 to 4, from 2 to 6, from 3 to 6 etc., aswell as individual numbers within that range, for example, 1, 2, 2.7, 3,4, 5, 5.3, and 6. This applies regardless of the breadth of the range.

Description

Described herein is a hemispheric breast imaging system (HABIS) thatacquires ultrasound scattering measurements using a hemispheric array oftransducers, and reconstructs the volume of a subject's breast using ahigh-performance computer network. As contemplated herein, there are twohardware components of HABIS, i.e., a data-acquisition apparatus and ahigh-performance computer network, have been integrated to implementefficiently and accurately an inverse scattering algorithm thatreconstructs high-resolution images of a subject's breast volume. In oneembodiment, the data-acquisition apparatus includes an array ofultrasound transducers arranged in a generally hemispheric pattern andall electronics required for transmitting and receiving ultrasoundsignals from the array. In one embodiment, the high-performance computernetwork includes a plurality of interconnected computer nodes configuredfor fast, parallel processing of the data received from thedata-acquisition apparatus.

Described herein is also an inverse scattering algorithm forreconstructing an image of a breast or other target scattering object.The general purpose of the inverse scattering algorithm is toreconstruct an image of the scattering object from measurements of theeffects that the object has on incident signals used to probe theobject.

However, reconstructions using existing techniques can be challengingfor a number of reasons. For example, the relationship between theobject and the scattering measurements is nonlinear, thus, high-qualityreconstructions generally require iterative procedures. Since only afinite set of scattering measurements can be acquired, the scatteringobject can never be completely characterized. The scattering informationis particularly limited when physical restrictions are present on therange of incident and receive signal directions. Further, fine detailsof the object correspond to small variations in the scatteringmeasurements, which can be overwhelmed by other scattering responses andcan be easily confused with system noise. Lastly, numerical methods thatameliorate these problems can be prohibitively time-consuming,computationally expensive, or both. HABIS resolves the above issuesthrough the implementation of a unique algorithmic and engineeringdesign as described herein.

As described herein, HABIS provides speckle-free, high-resolution,quantitative images of intrinsic tissue characteristics, e.g., soundspeed and attenuation slope, for improved detection, diagnosis, andtreatment monitoring of breast cancer. In one embodiment, the system canacquire data during an approximately two-second interval by using 10,240parallel channels for transmission and reception. The system can imagean entire breast volume within minutes, with isotropic point resolutionas good as the lateral resolution of x-ray mammography, by using analgorithm that independently and simultaneously reconstructs subvolumesspanning the breast. Using ultrasound, the system can be used to examinea breast with non-ionizing radiation for cancer detection. The systemovercomes limitations of x-ray mammography such as low resolution ofcontrast in dense breast tissue, i.e., breast tissue with high x-rayattenuation; distortion and discomfort resulting fromcompression-induced deformation of the breast, and poor imaging ofbreasts having implants. Accordingly, the system can significantlyimprove the detection and diagnosis of breast cancer, and can alsoimprove the monitoring of response to breast cancer treatment, comparedto systems currently available.

The reconstruction algorithm described herein can reconstruct subvolumesindependently, i.e., in parallel. Graphic processing units (GPUs) allowsmall high-performance computers (HPCs) to perform massively parallelcomputations that are ideally suited for implementation of the describedparallelized reconstruction algorithm. Accordingly, in one embodiment,the system can include a GPU-based HPC network coupled to adata-acquisition apparatus that enables reconstructions of the breastvolume to be obtained in a relatively short amount of time. For example,in one embodiment, the breast volume can be reconstructed in less than20 minutes, and in other embodiments, in less than 15 minutes, in lessthan 10 minutes, or in less than 5 minutes. However, it is contemplatedherein that a person skilled in the art could readily modify the systemto even further decrease the reconstruction time, for example by usingfaster processing units or by increasing the number of nodes. Incontrast to this, on a supercomputer using an existing method forreconstruction of a large single volume, such reconstructions canrequire time periods of as much as a few days. This improvement inreconstruction time provided by HABIS can have an enormous impact on theclinical utility of imaging systems because data can be acquired, andthe reconstructed volumes of the breast viewed, in the course of asingle visit. Accordingly, because HABIS is capable of producing imageswith at least 100 micron resolution in minutes, HABIS provides animproved and efficient way to screen for breast cancer and also todiagnose other breast diseases. In a manner previously unattainable.

Imaging System

In one embodiment, HABIS comprises an apparatus for acquiring data froman array of ultrasound transducers. In another embodiment, HABISincludes an examination table that houses or otherwise integrates atleast part or all of the data acquisition apparatus, which may includeboth its front-end and back-end electronics. In yet another embodiment,the data acquisition apparatus can be connected to a network ofcomputers that can process or otherwise perform the desired imagereconstruction. For example, in one embodiment, the front-endelectronics are arranged radially under the patient “head” end of thetable around the generally hemispheric transducer array. In oneembodiment, under the foot-end of the table are electronic modules thatcan include power supplies, power-supply filter boards, control boards,isolation transformers, high-voltage regulator boards, circuit breakers,and cable channels, as needed. In one embodiment, the system can alsoinclude other components suitable for use with a particular applicationassociated with the image acquisition, such as a fluid-handling cart andan operator console which can be positioned near the examination table.

Referring now to FIG. 1 , an exemplary embodiment of the dataacquisition apparatus 10 of the system is shown, wherein the apparatusis at least partially integrated into the surface of an examinationtable 20 generally having a surface 22 for a subject to lie on. Surface22 has an opening or cup 24 into which a scattering object, such as apatient's breast, can be placed. The hemispheric transducer array of thesystem may form the exposed walls of cup 24, or alternatively, cup 24may be a permanent or disposable container that fits within the cavityof the transducer array hemisphere to prevent direct contact of thescattering object with the transducers, and/or to hold a suitable amountof coupling fluid, gel or other material used during the imagingprocedure. For example, the container may be a detachable and disposablecup-shaped container. Accordingly, a patient can lie prone on tablesurface 22 with either the right breast or the left breast pendant incup 24. Further, cup 24 can be filled with a coupling fluid or any othersuitable material as desired during an ultrasound imaging procedure.FIG. 2 is a schematic of another embodiment of data acquisitionapparatus 10, wherein the examination table surface and a portion of thesides of table 20 have been removed. Again, the head end of table 20houses the hemispheric array assembly, which is designed to image anobject placed in cup 24.

FIG. 3 is another schematic of data acquisition apparatus 10, whereinall sides of table 20 have been removed to show a portion of ahemispherical array assembly 30. Hemispherical array assembly 30includes a plurality of transducers, or transducer element assemblies 31having an inner surface for transmitting and receiving ultrasoundsignals via a plurality of piezoelectric elements, and an outer surfaceconfigured with suitable circuitry for transferring electrical currentand signal data to/from the piezoelectric elements within the transducerfacets. Accordingly, the inner surface of cup 24 forms a cavity forholding a scattering object, for example a patient's breast, that can beimaged using ultrasound. Further, cup 24, or otherwise the cavity formedby the inner surfaces of the transducers, also defines the imagingregion within the apparatus or device. As contemplated herein, the cupcan comprise any material suitable for ultrasound analysis, as would beunderstood by a person skilled in the art. In the embodiments describedherein, the cup is substantially hemispherical in shape. However, thecup may be a shape other than a hemisphere, provided that the array ofultrasound transducers and other electronic components of the system,and also the image reconstruction algorithm described later herein, aresuitably modified to suitably image the resultant object volume withinthe imaging region.

In one embodiment, data acquisition apparatus 10 of the system canacquire a complete scan of over 100 million receive waveforms in about atwo-second time period, with minimal noise contributions to the receivewaveforms, and with very precise control over both the timing of thetransmissions and the sampling of the received waveforms. In variousembodiments, hemispheric array assembly 30 comprises a plurality ofdata-acquisition subassemblies connected to the transducer elements. Inone embodiment, hemispheric array assembly 30 comprises forty (40)data-acquisition subassemblies that are each connected to a singletransducer element assembly. However, the number of data-acquisitionsubassemblies is not limited to forty, and can be more or less thanforty, depending on the number of transducer assemblies and/or otherfactors, such as the desired resolution and/or speed of thereconstructed image.

Referring now to FIG. 4 , an exemplary embodiment of a transducerelement subassembly 31 is shown. FIG. 4A is a view of the inner surfaceof transducer element subassembly 31 comprising five triangular planarfacets 32. Each planar facet 32 comprises a plurality of piezoelectricelements that are located underneath the dimples or depressions on thesurface of facets 32. In FIG. 4B, a cross-sectional X-ray of anexemplary planar facet 32 is shown. Facet 32 includes a surface layer101 that comprises a plurality of depressions or dimples 102. In oneembodiment, a diverging lens is located in each dimple 102. In oneembodiment, surface layer 101 can be coated with a polymer, for examplea Parylene polymer. Beneath surface layer 101 are two matching layers105. In one embodiment, facet 32 can comprise a number of matchinglayers other than 2, for example 1 or 3 matching layers. Matching layers103 and 105 can comprise any material, and can be any thickness, thatwould be suitable for an ultrasound transducer matching layer, as wouldbe understood by a person skilled in the art. Beneath matching layers103 and 105 is a plurality of ultrasound transducer elements 107. Eachtransducer element 107 is substantially aligned with a dimple 102.Transducer elements 107 can comprise any material suitable for anultrasound transducer known in the art, for example PZT crystal.Further, facet 32 comprises a backing layer (not shown) beneathtransducer elements 107. The backing layer can comprise any suitablebacking material for an ultrasound transducer. In one embodiment, thebacking layer comprises a conductive epoxy. In one embodiment, facet 32can comprise additional layers, for example a conductive metal coatingon either side of the transducer elements, for grounding and excitation.In one embodiment, facet 32 can comprise additional components, forexample printed circuit boards. FIG. 4C is a view of the outer surfaceof transducer subassembly 31. Five triangular planar facets 32 areconnected together via a bracket 26. A plurality of wire connectors 110are located on the outer surface of each facet 32. Wire connectors 110can be used to communicatively couple transducer elements 107 to thedata-acquisition components described later herein, for example viaprinted circuit boards.

Referring now to FIG. 5 , transducer assemblies 31 of hemispheric arrayassembly 30 are shown coupled to forty front-end boards, i.e.,data-acquisition subassemblies 33 (only a portion of these boards areshown in FIG. 5 ). In this embodiment, data-acquisition subassemblies 33radiate from the axis of assembly 30. In one embodiment, each dataacquisition subassembly 33 comprises 256 channels of electronics, and isconnected to the transducer assemblies 31 via cable connections 34,which may be relatively short in length to minimize signal loss. In oneembodiment, cable connections 34 are PCIe cables. Hemispheric arrayassembly 30 also includes bottom-plane boards 35 for distributing powerand timing, as would be understood by those skilled in the art. In oneembodiment, data collected by at least one of the 256-channel dataacquisition subassemblies 33 can be sent to imaging computers throughsuitable cables, e.g., PCIe cables. In one embodiment, the cables may beabout five meters long, but it should be appreciated that these cablescan be any length desired, as would be understood by a person skilled inthe art. This arrangement allows each ultrasound transducer elementassembly 31 to be in close proximity with the front-end circuitry thatis used to collect signals from the element. The close proximity betweenthe transducer elements and the data acquisition circuitry lowers signalloss and extends the range and angle over which useful signals can betransmitted and received.

Referring now to FIG. 6 , the forty sets of data-acquisition electronicssubassemblies 33 are shown mounted in groups of eight on five bottomplane boards 35 that together form an annulus encircling the axis of thehemisphere array in a plane below the hemisphere. Bottom-plane boards 35are used to distribute timing and control information as well as toprovide power to the forty data-acquisition subassemblies 33. Thisdata-acquisition subassembly and bottom-plane arrangement enableshigh-precision timing signals to be distributed without appreciabledegradation. In the embodiment shown, each data-acquisition subassembly33 includes four blocks of 64-channel electronics positioned insuccessive angular sections of the board. This arrangement allowstransducer assemblies 31, comprising a total of 10,240 transducerelements, to be in proximity to the hemispheric array electronics. Insuch an embodiment, transducer assemblies 31 are connected to theelectronics through 160 short cables.

FIG. 7 shows how the 256-channel data-acquisition subassemblies 33 pluginto hemispheric array assembly 30 and shows the conical frame 39 thatsupports the inboard edge of the 256-channel subassemblies 33. Conicalframe 39 also forms a plenum to direct air flow, for example air flowfrom fans that can be placed in another portion of the examinationtable. In one embodiment, each transducer element subassembly 31includes a bracket 26. In one embodiment, brackets 26 can be connectedto data-acquisition subassemblies 33 via a plurality of flexiblesupports 27. Further, transducer element assemblies 31 are connected todata-acquisition subassemblies 33 via wiring suitable for sending andreceiving signals (not shown).

In one embodiment, during each two-second scan, every data acquisitionsubassembly 33 receives 10,240 waveforms with 4,096 temporal samples ineach waveform from 256 different transducer elements located intransducer assemblies 31. Thus, a total of 32.2 GB is acquired by eachsubassembly. By comparison, if a typical computational system would beused for reconstruction, then the data from all forty data acquisitionsubassemblies 33, i.e., a total of 40×32.2 GB, or 1.28 TB, would need tobe aggregated and transferred to that system. However, a significantpart of the processing can be performed on data from each acquisitionsubassembly 33 without reference to data from the other acquisitionsubassemblies. As a result, the amount of data that needs to betransferred to a computer system to complete the reconstruction, afterthe initial phase of independent parallel processing has been completed,is about 1,000 times smaller than without the use of such parallelprocessing. Currently available GPU architectures have the processingcapability to complete the computations required for HABIS. Accordingly,data acquisition apparatus 10 includes GPU cards that are distributed inthe nodes of the data acquisition network. Further, superimposing a webof InfiniBand connections on the network enables rapid inter-nodecommunication necessary in later stages of the computation so that theremainder of the reconstruction computations can be completed withsimilar speed. As a cost-saving and space-saving measure, in oneembodiment, the HABIS data-acquisition subassemblies each incorporate256 channels of circuitry. Such an arrangement can be used to preventnoise from the digital electronics from interfering with analogreception.

FIG. 8 is a diagram showing the power supply assembly 40 positionedwithin the interior of table 20. Power supply assembly 40 can includecomponents necessary for the operation of hemispheric array assembly 30,including, but not limited to: a plurality of power supplies 41,power-supply filter boards 42, isolation transformers 44, cooling fans46, air ducts 48, high-voltage regulator boards, a power-control module,wiring connection blocks, disconnect switches, and circuit breakerswithin examination table 20. Alternatively, in one embodiment, suchcomponents can be placed in a location outside of examination table 20,for example in a secondary cabinet.

FIG. 9 is a diagram showing examination table 20 in an examination room.Examination table 20 can be connected to an operator console 51, whichcan be used to control any portion of the system, as needed. Inaddition, a fluids cart 50 can be connected to table 20 to provide asupply and/or recirculation of coupling, i.e., ambient fluid to cup 24.Further, table 20 can be connected to a computer network/scalable serverset 70 for image reconstruction, as will be described later herein.

In various embodiments, the components of HABIS can be connected to oneor more power supplies suitable to provide power for their operation.For example, in one embodiment, examination table 20 can be connected toa three-phase 208-volt AC supply through a disconnect switch and anisolation transformer. Computer network 70 can be connected to anotherthree-phase 208-volt AC supply through another disconnect switch. In oneembodiment, no isolation transformer is included in computer network 70because the only connection between examination table 20 and computernetwork 70 is a set of 40 low-voltage PCIe cables. However, the variouscomponents of HABIS can be supplied power in any manner as would beunderstood by a person skilled in the art, and the power supplyarrangement is not limited to the specific embodiments described herein.

As previously described, in one embodiment, HABIS is an integrateddata-acquisition apparatus and high-performance network that rapidlycollects imaging data and efficiently implements fast parallelcomputation of images via an inverse scattering algorithm. In such anembodiment, the architecture of the system permits GPU-based parallelcomputations on each node and InfiniBand-based aggregation of results ateach node. The design of the system avoids extensive time-consuming dataaggregation, permits parallel computation of refined data sets thatsubstantially reduce the amount of data transferred between computernodes for subsequent parallel reconstruction of subvolumes, and enablesfast transfer of intermediate computational results between computingnodes. The parallel computation takes place on two levels: a high levelon each computing node and a low level on each GPU. A head node in thenetwork can provide command, control, and monitoring. In one embodiment,an Internet connection can enable command, control, and monitoring to beperformed remotely. Data-acquisition apparatus 10, as shown anddescribed herein, provides a large number of independent channels, whichcan accommodate a large volume of data in a short time period, therebyexceeding the performance of currently available systems. For exampleand without limitation, the system architecture may include separateconnections between pairs of data-acquisition electronics sets and nodesof the high-performance computer network. These connections allowprocessing to take place prior to aggregation of the receive signals.The system architecture may also include configuration of individualcompute nodes with cost-efficient resources that efficiently perform theparallel computations used in HABIS reconstructions. The systemarchitecture may also include InfiniBand connections and switching thatefficiently perform in a cost-effective manner the burst transfers ofdata after each stage of computation. The system architecture may alsoinclude a mechanical configuration of the data-acquisition apparatus tominimize the lengths of the data paths between the transducer elementsand the transmit and receive electronics so that the range and angleover which useful signals can be transmitted and received is greatlyextended. The system architecture may also include circuitry designs toreduce cost by consolidation of data-acquisition and data processingelectronics. Further, the system architecture may also include a timingand control system that meets the stringent tolerances imposed bycoherent imaging and transmission encoding.

Transducer Elements and the Hemispheric Array

In one embodiment, the array of transducer elements of HABIS is arrangedin a faceted construction to provide imaging of a generally hemisphericvolume. As shown in FIG. 10 , in one embodiment, the hemispheric arrayin HABIS is approximated using 40 triangular planar facets. In one suchembodiment, ten of the facets are equilateral triangles and 30 of thefacets are isosceles triangles. The use of a faceted approximation of ahemisphere with two types of triangular planar elements greatlysimplifies the construction of the array. Further, in such anembodiment, each facet contains 256 small diameter piezoelectricelements, as indicated by the dots on the facets in FIG. 10A, made froma ceramic material composite that suppresses undesirable lateral modesof vibration.

In one embodiment, the elements on the facets are positionedpseudorandomly. The pseudorandom positions permit the use of many fewerelements (and, thus, fewer independent transmit and receive channels)than would be otherwise required to avoid grating lobes during theformation of transmit and receive beams. In one embodiment, oneconfiguration of pseudorandom positions is used on the equilateralfacets and another configuration of positions is used on the isoscelesfacets. The use of two configuration sets of pseudorandom positionssignificantly facilitates practical fabrication of the array withoutdegrading the capability of the array to form beams.

Each element may include a diverging lens that broadens the pattern ofthe transmit beam and the pattern of the receive sensitivity. Thisbroadening extends the volume, i.e., the solid angle, of the scatteringobject illuminated by transmissions from the elements. The broadeningalso extends the volume i.e., the solid angle, of the scattering objectfrom which ultrasound signals can be received by the elements. Theresult of the extended coverage is an appreciably enhanced capability toconcentrate focuses formed using the array.

Each element in the array may include two matching layers. The matchinglayers produce a wider temporal-frequency bandwidth than would beobtained without the layers. The matching layers also increase thetransmission energy over the energy that would be transmitted withoutthe layers. Additionally, the layers increase the sensitivity ofreception over the reception sensitivity that would be obtained withoutthe layers. Further, the transmit waveform applied to the transducerelements is designed to concentrate the transmitted energy within thetemporal-frequency bandwidth of elements.

In addition, the receive circuitry contains a tuning element, i.e., aninductor, that cancels the bandwidth-narrowing effect of the capacitanceassociated with the transducer elements, the cable between thetransducer element and the front-end electronics, the parasiticcapacitance of the printed circuit wiring, and the input capacitance ofthe low-noise amplifier in the receiver chain.

The use of transmissions that are coded without degrading crosstalk andthe ability to decode receptions without degrading crosstalk is enabledby the front-end electronics assembly design of HABIS. In the system,the connection between the transducer elements and the cable to theelectronics is made through a special signal redistribution arrangementimplemented on a printed circuit board with wide trace separation andshielding. Additionally, a special pattern of shielded connectionsbetween the cable from the transducer and printed circuit boardcontaining the front-end electronics reduces crosstalk significantlycompared to the crosstalk that would otherwise exit. The tuning elementsnoted above are shielded and widely separated to suppress magneticcoupling between channels.

High-Performance Computer Network

As previously described, in one embodiment, HABIS includes ahigh-performance computer network that is connected to thedata-acquisition assembly. A specialized architecture consisting of adata-acquisition apparatus and a high-performance computer network isrequired to collect the ultrasound data in a short time and toreconstruct the breast volume in minutes thereafter. This is because acomputational capability must be integrated into HABIS to enableimmediate parallel preprocessing of the acquired data. The initialcomputations performed in the data-acquisition subassemblies yieldrefined data sets that substantially reduce the amount of datatransferred between computer nodes for subsequent parallelreconstruction of subvolumes. As mentioned elsewhere herein, the systemarchitecture may include 40 sets of data-acquisition subassemblies, witheach set containing 256 independent transmit and receive channels, and20 high-performance computing nodes with each node containing four GPUs.In other embodiments, comparable architectures with different numbers ofdata-acquisition subassemblies and high-performance computing nodes canbe used, as the system is not limited to the specific numbers andarrangements of data-acquisition subassemblies and computer nodesdescribed herein.

FIG. 11A is a diagram of an exemplary embodiment of a scalable computingnetwork suitable to meet the processing needs for HABIS as describedherein. For example, FIG. 11A shows the interconnections of a 21-nodenetwork that is comprised of 20 hybrid (acquisition and computation)nodes and a head (or control) node, which supervises the computingnetwork. In this embodiment, the 40 data-acquisition boards are arrangedin five groups of eight. Two data-acquisition boards are connected toeach of the 20 computing nodes. The head node may also serve as aconsole for rendering images and for controlling the instrument. FIG.11B shows a group of eight boards and their corresponding nodes ingreater detail. The 20 computing nodes, each with two GPUs, form anefficient numerical engine for massively parallel computations.Single-precision GPUs are ideal computational engines for severalreasons. Since the data acquired from the hemispheric transducer have 14bits of precision, and since the computations applied to this data arenot numerically unstable, single-precision arithmetic is sufficient. Infact, reduced memory requirements and faster transfer rates associatedwith single-precision data make this format desirable. Eachreconstruction requires a very large number (about 50×10¹⁵floating-point operations, i.e., 50 petaflops) of single-precisionoperations. Each node receives data from two of the 40 sets ofdata-acquisition electronics (shown as the green rectangles that containfour 64-channel blocks in FIG. 11B) via fast PCIe connections. Four GPUsare also connected to each node to form a powerful single-precisioncomputational engine for parallel processing. InfiniBand connections toa central switch enable rapid inter-node communication necessary betweenstages of the computation. These switches allow 20 InfiniBandconnections to be active simultaneously. In one embodiment, all thenodes of the network can also be connected to a local Ethernet switch(not shown) for command, control, and monitoring. A high-speedconnection of the local Ethernet to the Internet further enablescommand, control, and monitoring from a remote location.

The reconstruction algorithm, which is described in detail later herein,can be parallelized at both a high and a low level. The high-levelparallelization factors the computations into large-scale operationsthat are performed independently on separate nodes. The low-levelparallelization reduces these operations to a succession of vector andmatrix computations that can be implemented as GPU kernels. Also, only amodest amount of logic is needed to direct the flow of data and tosequence the computations. Careful sequencing of these kernels can usethe asynchronous data transfer capability of currently availablegeneral-purpose GPUs to eliminate data transfer latencies, resulting inimproved efficiency. Data-transfer latencies are further suppressed bygeneration 3 PCIe connections used in currently availablegeneral-purpose GPUs.

During an approximately two-second period of data acquisition thatcomprises a volume scan by HABIS, each of the 40 sets ofdata-acquisition electronics subassemblies receives 10,240 waveformsfrom 256 different transducer elements. For a sampling rate of 20 MHzover a time interval of 204.8 μs and for two bytes per sample, all ofwhich are representative of those used in HABIS, a total of (2 bytes persample)×(20 samples per s)×(204.8 μs per waveform)×(10,240 waveforms perelement)×(256 elements)=21.47483×10¹⁰ bytes or about 21.5 gigabytes (GB)are acquired and stored by each set of electronics. As mentionedpreviously, if a separate computational facility were used forreconstruction, then the data from all 40 sets of the data-acquisitionelectronics (i.e., a total of 40×21.47483×1010 bytes=8.589934×10¹¹ bytesor about 0.859 TB) would need to be aggregated and transferred to thatfacility, a time-consuming process given the large amount of data to betransferred.

As contemplated herein, FIG. 12 illustrates an exemplary data processingflow (via the image reconstruction algorithm) that permits processing tobe performed in stages that consist of parallel computations on eachnode. As shown, these parallel computations on one node may becompletely independent of the computations on the other nodes. As shownin FIG. 12 , Stage I operations include preprocessing steps (such astime-variable gain compensation, transmission decoding, and applicationtime gates to isolate the response from different regions), b-scanimaging, and initial approximation of the scattering object. Stage IIoperations include split-step propagation of fields that emanate fromeach of the 10,240 transducer elements and pass through the approximatescattering object, reduction of these fields to simplifiedrepresentations each limited to a single subvolume, and estimation ofthe portions of the scattering measurements attributable to theapproximate scattering object. Stage III operations produce a refinedapproximation of the scattering object by using residual scatteringmeasurements to reconstruct residual medium variations in eachsubvolume. After each stage of computations, the results of thecomputations at each node are redistributed to other nodes for the nextstage of computations.

Computational efficiency and cost-effective implementation of the stagedprocessing scheme described above were the main criteria used to designthe high-performance computer network. Two key requirements follow fromthese criteria. One requirement is that the nodes must have enoughcomputational capacity to complete each stage of computation in a timelymanner. The other requirement is that the nodes must be able to exchangedata rapidly enough so that data redistribution is not a bottleneck.

The computer nodes must also be endowed with enough memory to insurethat the GPUs can be continuously supplied with input data, and alsohave enough storage for intermediate results so that saving andrecalling data from disk files is unnecessary. In one embodiment, eachHABIS compute node is equipped with four GPUs and a total of 128 GB ofmemory. These allocations allow each node to conduct four completelyindependent parallel computations during each processing stage. Thiscombination of resources results in relatively low-cost compute nodesthat have precisely the right kind of numerical capability for efficientimplementation of the HABIS algorithm. In other embodiments, each HABIScompute node can include a different number of GPUs and/or memory, aswould be understood by a person skilled in the art.

The transfer of data from the acquisition apparatus to the computernetwork and the node-to-node data transfers that occur after each stageof computation all involve large volumes of data. Each compute node inthe architecture is responsible for collecting a receive signal of 4,096two-byte temporal samples from a group of 512 receive channels for eachof 10,240 transmissions. Based on the previously-noted volume of datafor a 256-channel set, the total volume of all these samples is about 43GB. Excluding overhead, this volume of data can be transferred over afast 16-lane PCIe connection with an overall transfer rate of 8 GB/s inabout 5.4 seconds and can be transferred over a pair of such PCIeconnections in about 2.7 seconds.

Preprocessing occurs initially in each node without internodecommunication and consists of transmit-receive channel responseequalization, compensation for time-varying gain, Fouriertransformation, and signal decoding requires expansion of the two-bytesamples to a larger size. In HABIS, although other expansions arepossible, the expansion to a larger size is accomplished by conversionof the two-byte integer samples into four-byte floating-point values.This increases the volume of data associated with one node to 86 GB from43 GB. Although Fourier transformation that is part of the preprocessingexpands the four-byte floating-point values to complex values eachoccupying eight bytes, the volume of the complex data is orders ofmagnitude less if only a set of Fourier components in the usefulbandwidth of the system are retained and the relation between positiveand negative frequencies imposed by real temporal signals is used. Inview of this, node-to-node data transfers that proceed subsequentprocessing involve smaller volumes of data.

The required node-to-node transfers can be performed quickly usingInfiniBand connections instead of PCIe connections. The efficiency ofInfiniBand connections is illustrated by considering the transfer ofresults from the parallel Stage II calculations, i.e., thescattering-measurement matrix {tilde over (M)} with entries {tilde over(M)}_(mn) for the approximate scattering object. This matrix iscomprised 10,240×10,240 complex numbers that are each eight bytes. Thus,the size of the entire matrix is about 0.84 GB, of which one-twentiethresides on each compute node. The aggregation of the entire matrix ateach node for the Stage III computations requires a total transfer of

$\begin{matrix}{{\underset{\underset{\begin{matrix}{{Nodes}\mspace{14mu}{that}\mspace{14mu}{contain}} \\{a\mspace{14mu}{portion}\mspace{14mu}{of}\mspace{14mu}{\overset{\sim}{M}}_{nm}}\end{matrix}}{︷}}{\underset{︸}{20}} \times \underset{\underset{\begin{matrix}{{Size}\mspace{14mu}{of}\mspace{14mu}{each}} \\{{portion}\mspace{14mu}{of}\mspace{14mu}{\overset{\sim}{M}}_{nm}}\end{matrix}}{︷}}{\underset{︸}{0.84\mspace{14mu}{{GB}/20}}} \times \underset{\underset{\begin{matrix}{{Nodes}{\;\mspace{11mu}}{to}\mspace{14mu}{get}} \\{{{each}{\;\mspace{11mu}}{portion}}\mspace{14mu}}\end{matrix}}{︷}}{\underset{︸}{19}}} = {16\mspace{20mu}{{GB}.}}} & {{Eq}.\mspace{11mu}(30)}\end{matrix}$

For a 4X QDR InfiniBand connection that, excluding overhead, has a datatransfer rate of about 4 GB/s (i.e., 32 Gb/s), this transfer can becompleted in four seconds using a single InfiniBand connection, and canbe completed in about 0.2 seconds using 20 simultaneously-activeInfiniBand connections.

Communication and Control

The preprocessing and b-scan imaging that take place in Stage I areperformed with no communication between the 20 nodes in the network.However, full inverse scattering reconstruction is completed by theStage II and Stage III computations that require node-to-node transfersof data over the InfiniBand network. The 20 nodes also need to receivecommands from an operator using a terminal to control the datacollection, processing operations, b-scan image formation, and imagereconstruction via inverse scattering. Communication software providesthe necessary capabilities for data exchange, distributed control, andcoordinated calculations among the nodes. Each of the 20 nodes includesa library that provides an interface to the other 19 nodes and a headnode with a keyboard and monitor. From the head node, commandsoriginated by an operator can be broadcast to the 20 computing nodes byusing control program that runs on each node. The program can includelogic that allows each node to determine its data-acquisition functionsfrom the operator-originated requests. The program also supervises thepreprocessing calculations. Additional programs that run on all thenodes form b-scan images and reconstruct images via inverse scattering.The terminal that accepts operator commands can be used as a userinterface that runs on one of the 20 nodes or, alternatively, on thehead node.

HABIS Algorithm

Inverse Scattering Framework

Monochromatic, i.e., single-frequency, measurements of scatteringmeasurements are used in the HABIS imaging algorithm. The measurementsare conveniently arranged as a two-dimensional matrix M of complexvalues with 10,240 rows and 10,240 columns. The value of the matrixentry M_(mn) with row index m and column index n is the scatteringamplitude that is measured at the n-th receiving element in response toa transmission from the m-th element.

Scattering objects are reconstructed by determining variations of thebackground medium Δη(x) that account for the observed scatteringmeasurements. This determination requires a mathematical model thatrelates the medium variations to the scattering measurements. Aweak-scattering model, i.e., one in which the scattered wave is smallrelative to the incident wave, is desirable because the scatteringmeasurements in a weak-scattering model depend linearly on the mediumvariations. The explicit expression for weak scattering isM _(mn)=∫ψ_(n)(x)ψ_(m)(x)Δη(x)d ³ x  Eq. (1)where ψ_(m)(x) denotes the spatially varying amplitude of the field thatis transmitted by element m and propagates through an empty background,and ψ_(n)(x) denotes the spatially varying sensitivity pattern ofreceiver n that also propagates through an empty background. Thetransmit field of element n is the same as the receive sensitivitypattern of element n.

Unfortunately, weak scattering is usually not very accurate in practicalapplications because, in actual measurements, the value ψ_(m)(x) of thetransmit field that appears in the integrand of Eq. (1) is altered bymedium variations encountered between the transmit element and x, and,likewise, the value ψ_(n)(x) of the receiver sensitivity pattern thatalso appears in the integrand of Eq. (1) is altered by medium variationsencountered between x and the receive element. A more accurate model ofthe scattering measurements is formulated by incorporating anapproximate scattering object in the scattering expression. This yieldsthe revised scattering expressionM _(mn) −{tilde over (M)} _(mn)=∫ψ_(n)(x){tilde over (ψ)}_(m)(x)Δη(x)d ³x  Eq. (2)where {tilde over (ψ)}_(m)(x) denotes the spatially varying amplitude ofthe field transmitted from element m that propagates through theapproximate object, ψ_(n)(x) denotes the spatially varying sensitivitypattern of receiver n that propagates through the approximate object,{tilde over (M)}_(mn) denotes the scattering from the approximateobject, and Δη(x) denotes medium variations that are relative to theapproximate object.

The integral on the right side of Eq. (2) is a distorted weak scatteringexpression for the residual scattering measurement M_(mn)−{tilde over(M)}_(mn). Equation (2) reduces to the original weak scatteringexpression given by Eq. (1) when an empty background is used for theapproximate scattering object, but Eq. (2) is more accurate than Eq. (1)when the approximate object is a good approximation of the true object.For this reason, inverse scattering algorithms are usually iterative. Ineach iteration, the approximate scattering object ηq(x) is improved byappending a refinement term Δη(x) that is derived from Eq. (2). This isthe general framework for inverse scattering. The main differencesbetween specific implementations are in the way that the refinements areestimated.

An important feature of these iterative procedures in breast imagingapplications is that the quality of refinements to the approximatescattering object does not improve uniformly. The reason for this can befound in Eq. (2) and does not depend on the way that Δη(x) is estimated.In Eq. (2), the contribution to the residual scattering measurementM_(mn)−{tilde over (M)}_(mn) that is due to the medium variation Δη(x)at x is weighted by the factor {tilde over (ψ)}_(n)(x) {tilde over(ψ)}_(m)(x). However, {tilde over (ψ)}_(n)(x) and {tilde over(ψ)}_(m)(x) are approximate oscillatory signals that do not take intoaccount the influence of variations to the approximate object that areencountered as the fields propagate to x from the transmit element, andfrom x to the receive element. If these influences cause appreciablephase shifts in {tilde over (ψ)}_(n)(x) and {tilde over (ψ)}_(m)(x),then the scattering contribution that Eq. (2) attributes to Δη(x) willbe weighted erroneously. Thus, Eq. (2) is only an accurate expression ofthe relationship between the scattering measurements and the mediumvariations in regions where the field approximations {tilde over(ψ)}_(n)(x) and {tilde over (ψ)}_(m)(x) are accurate, and any estimateof Δη(x) that is derived from Eq. (2) can only be expected to improvethe approximation of the scattering in these regions. Furthermore, tomake significant improvements in the accuracy of the fieldapproximations a global update of the approximate scattering object isneeded that refines the medium variations along the full range ofpropagation paths that emanate from the transducer elements. Thesecircumstances result in very slow convergence of the iterativerefinements described above when the true scattering object is poorlyapproximated. However, much more rapid convergence can be expected whenthe approximate scattering object is close enough to the true object toallow accurate representation of transmit fields and receiversensitivity patterns.

The HABIS imaging algorithm substitutes a two-step procedure for thegeneral iterative scheme outlined above. First, an approximatescattering object is estimated to obtain field approximations that aregood enough to ensure that Eq. (2) is accurate in the region ofinterest. A single inverse-scattering refinement is then computed toproduce a sharper and more detailed reconstruction of the mediumvariations in this region. Starting with a high-quality initial estimatecan bypass many iterations in the early stage when convergence is slow.The method used to obtain the initial approximation to the scatteringobject and the method used for iterative refinement of the scatteringobject are described in the sections below along with other importantaspects of the HABIS imaging methodology.

Although formation of the initial approximation logically precedesiterative refinement, the method for iterative refinement is discussedfirst as this method is more fundamental.

Refinement of an Approximate Scattering Object

In this section, the following quantities are assumed to be known:

η(x): medium variations that constitute an approximate scattering object

{tilde over (ψ)}_(m)(x): spatially varying amplitude of fieldtransmitted from element m that propagates through η(x)

{tilde over (ψ)}_(n)(x): spatially varying sensitivity pattern ofreceiver n that propagates through the approximate object

{tilde over (M)}_(mn)(x): computed scattering from η(x) that would bereceived at element n in response to a transmission from element m, (m,n=1, 2, . . . , 10,240)

M_(mn): measured scattering that is received from the true object atelement n in response to a transmission from element m (m, n=1, 2, . . ., 10,240)

The above list includes all the terms that appear in Eq. (2) except forthe unknown variation Δη(x). The methods that are used to estimate Δη(x)from Eq. (2) are described below.

First estimate of Δη(x)

The typical strategy for estimation of Δη(x) is to adopt afinite-dimensional representation of Δη(x) that reduces Eq. (2) to asystem of linear equations. Some commonly used parameterizations ofΔη(x) include: sample values at the vertices of a rectilinear grid, alinear combination of sincfunctions that are centered at the vertices ofa rectilinear grid, or a linear combination of the eigenfunctions of ascattering operator. The systems of equations that result from theseparameterizations are usually either underdetermined or overdeterminedand must be supplemented with auxiliary conditions, such as normminimization, to ensure that the solution is unique andwell-conditioned. In many cases, the numerical problems associated withsolving these large systems of equations are significant.

The approach used by the HABIS algorithm avoids these time-consumingnumerical challenges. The value of Δη(x) at each location is assumed todepend linearly on the residual scattering measurements and may,therefore, be estimated by

$\begin{matrix}{{(x)} = {\sum\limits_{n,{m = 1}}^{10,240}{{\alpha_{n\; m}(x)}( {M_{mn} - {\overset{\sim}{M}}_{mn}} )}}} & {{Eq}.\mspace{11mu}(3)}\end{matrix}$where α_(nm)(x) is a matrix of coefficients that varies from point topoint. To determine these coefficients the expression on the right sideof Eq. (2) is substituted for the residual scattering measurementM_(mn)−{tilde over (M)}_(mn)(x) that appears on the right side of Eq.(3). Exchanging the order of the sum and the integral gives

$\begin{matrix}{{(x)} = {\int{\lbrack {\sum\limits_{n,{m = 1}}^{10,240}{{\alpha_{n\; m}(x)}{{\overset{\sim}{\psi}}_{n}( x^{\prime} )}{{\overset{\sim}{\psi}}_{m}( x^{\prime} )}}} \rbrack\Delta\;{\eta( x^{\prime} )}d^{3}x^{\prime}}}} & {{Eq}.\mspace{11mu}(4)}\end{matrix}$

The quality of this estimate depends on how closely the bracketed termin Eq. (4) approximates the translated δ-function δ (x′−x). A naturalway to realize this approximation is to set α_(nm)(x)=β_(n)(x) β_(m)(x),where the p factors are chosen to form a synthetic focus of the transmitand receive fields at x. For example, if δ_(m)(x)=1/{tilde over(ψ)}_(m)(x) and β_(n)(x)=1/{tilde over (ψ)}_(n)(x), then the bracketedsum in Eq. (4) is equal to

$\lbrack {\sum\limits_{m = 1}^{10,240}\frac{{\overset{\sim}{\psi}}_{m}( x^{\prime} )}{{\overset{\sim}{\psi}}_{m}(x)}} \rbrack^{2}$The value of this double sum is (10,240)² when x′=x. At other locations,however, the magnitude of the sum is much smaller because at otherlocations the summands do not accumulate coherently. Thus, the bracketedterm in Eq. (4) is concentrated at values of x′ that are near x whenα_(nm)(x)=β_(n)(x) β_(m)(x).Improvement of the Estimate of Δη(x)

Several adaptations of the estimate in Eq. (3) are needed, however, tonormalize and to improve the accuracy of the estimate. These adaptationsare facilitated by subdivision of the scattering object into overlappingsubvolumes. Refinements of the approximate scattering object in eachsubvolume are estimated independently. These refinements are thenblended together in the regions where the subvolumes overlap. For therest of this section, the procedures for estimating Δη(x) are intendedfor application to variations within a single rectangular subvolume.Also, for the rest of this section, the origin of the spatial coordinatesystem is assumed to be the center of the subvolume.

Setting α_(nm)(x)=β_(n)(x) β_(m)(x) yields, for the bracketed term inEq. (4), the expression

$\begin{matrix}{{K( {x,x^{\prime}} )} = {\sum\limits_{n,{m = 1}}^{10,240}{{\beta_{n}(x)}{\beta_{m}(x)}{{\overset{\sim}{\psi}}_{n}( x^{\prime} )}{{\overset{\sim}{\psi}}_{m}( x^{\prime} )}}}} & {{Eq}.\mspace{11mu}(5)}\end{matrix}$where K (x, x′) may be interpreted as the imaging kernel for theestimate

$\begin{matrix}{{(x)} = {\sum\limits_{n,{m = 1}}^{{10},{240}}{{\beta_{n}(x)}{\beta_{m}(x)}{( {M_{mn} - {\overset{\sim}{M}}_{mn}} ).}}}} & {{Eq}.\mspace{11mu}(6)}\end{matrix}$The kernel given by Eq. (5) is not translation invariant, but is nearlytranslation invariant in local neighborhoods. Thus, for sufficientlysmall subvolumes Eq. (4) can be rewritten

$\begin{matrix}\begin{matrix}{{(x)} = {\int{{K( {x,{x^{\prime} - x}} )}\Delta\;{\eta( x^{\prime} )}d^{3}x^{\prime}}}} \\{= {\int{{K( {{x^{\prime} - x},0} )}\Delta\;{\eta( x^{\prime} )}d^{3}{x^{\prime}.}}}}\end{matrix} & {{Eq}.\mspace{11mu}(7)}\end{matrix}$This equation expresses the estimate

(x) as the convolution of Δη(x) with the impulse response

$\begin{matrix}{{K( {x,0} )} = {\sum\limits_{n,{m = 1}}^{10,240}{{\beta_{n}(x)}{\beta_{m}(x)}{{\overset{\sim}{\psi}}_{n}(0)}{{{\overset{\sim}{\psi}}_{m}(0)}.}}}} & {{Eq}.\mspace{11mu}(8)}\end{matrix}$

The representation of

(x) given by Eq. (7) implies that a more accurate and normalizedestimate,

(x), can be derived by deconvolution of K (x, 0) from

(x). However, this deconvolution can only reconstruct spatial-frequencycomponents of Δη(x) at the spatial frequencies where the Fouriertransform of K (x, 0) is appreciable. This range of usablespatial-frequency components is constrained by the boundaries of thesolid angle of incident directions of the transmit fields (and,equivalently, by the receiver sensitivity patterns) that illuminate thesubvolume, and also by the temporal-frequency of the scatteringmeasurements. For subvolumes that are contained in the interior of thehemisphere, these limitations confine the usable spatial frequencies toan asymmetric region that includes the spatial-frequency origin.

Further improvement of

(x) is realized by extending the set of usable spatial frequencies to asymmetric region that contains the original asymmetric region. Theimprovement is accomplished by using low-spatial-frequency components of

(x) to model the relationship between the real and imaginary parts ofΔη(x). This relationship, together with symmetry properties of theFourier transform, can then be used to evaluate spatial-frequencycomponents of Δη(x) at spatial frequencies that are antipodal to thespatial frequencies in the original asymmetric region. This results inthe final estimate

(x) of the Δη(x) that is used to add detail and resolution to theapproximate scattering object.

Accordingly, important aspects of the procedure described above that aredifferent from other inverse scattering methods include: 1) The HABISprocedure for estimating Δη(x) separates the overall estimate intoestimates for each of the subvolumes that can be computed independently.The computations are, therefore, ideally suited to high-levelparallelization that assigns estimates of each subvolume to a differentcompute node of a high-performance computer network; 2) The estimate ineach subvolume is found by direct computations (i.e., computations thatare not directed by conditions that depend on results of priorcomputations) and that can be ‘vectorized,’ i.e., executed in parallel.This is the ideal setting for the kind of low-level parallelization thatis provided by numerical acceleration engines such as graphicsprocessing units known as GPUs; and 3) The initial estimate

(x) that is obtained from synthetic focusing is not as accurate asestimates that are derived from more costly algorithms, but the improvedestimates

(x) and

(x) are able to restore full accuracy with only a fractional increase incomputation.

Initial Approximation of the Scattering Object

An initial approximation of the scattering object is found by imagingmethods that are similar to b-scan algorithms used in currentlyavailable commercial ultrasound imaging systems. As noted in thepreceding section, initial estimates with good accuracy are needed toeliminate slowly converging iterations. These estimates are developedfrom a scattering model that is similar to the weak scattering model ofEq. (1), but with an important difference. The scattering measurementsused to form the initial estimate are temporal responses to a pulsetransmission while the scattering measurements represented by Eq. (1)are amplitudes of harmonic responses to monochromatic transmissions.

For efficient computation, the medium variations are estimated insubvolumes that can be reconstructed in parallel. However, in this case,the interior of the hemisphere is partitioned into tetrahedrons ratherthan cubes. Each tetrahedron has one vertex at the origin of thehemisphere, and three other vertices that form a subtriangle of one ofthe triangular facets of the hemispheric array. FIG. 10 shows thetetrahedral regions that are associated with subtriangle A of thehemispheric array. Since the hemispheric array previously describedherein consists of 40 facets and each facet is divided into foursubtriangles, the interior of the hemisphere is partitioned into 160tetrahedrons.

The medium variations in each tetrahedron are estimated from thescattered signals that are received by the 64 transducer elementscontained in the subtriangle of the hemispheric array that forms theouter face of the tetrahedron. Since the data-acquisition apparatustransfers all the received signals from the elements in this subtriangleto the same compute node, the signals are automatically distributed indata sets that are suitable for this parallel computation. Each receiveelement collects a total of 10,240 signals. However, only 256 of thesesignals are used in the estimate of the medium variations in thetetrahedron that is bounded by subtriangle A because the estimate ofthis subvolume only uses signals from transmit elements that belongeither to subtriangle A or belong to one of the three subtriangles A¹,A², A³ that have a common edge with subtriangle A.

Imaging Algorithm

To derive a mathematical scattering expression for the b-scan model, letp(t) denote the incident pulse that is used to excite the transmitelements that is centered at t=0. If directional effects and aberrationare neglected, then the pressure variations at x that result from thetransmit element at x_(m) are given by p (t−∥x−x_(m)∥/c₀)/∥x−x_(m)∥,where c₀ is the average speed of sound in the medium. Reflections ofthis incident wave that are caused by an inhomogeneity η(x) at x aredetected by the receiver element at x_(n) as the signal.

$\begin{matrix}{\frac{p( {t - {\delta_{nm}(x)}} )}{{{x - x_{m}}}{{x - x_{n}}}}{\eta(x)}} & {{Eq}.\mspace{11mu}(9)}\end{matrix}$where δ_(nm) (x) denotes the geometric timedelay(∥x−x_(m)∥+∥x−x_(n)∥/c₀. Since reflections can come from anyilluminated inhomogeneity, the total reflected pressure detected atx_(n) is given by

$\begin{matrix}{{\varphi_{mn}(t)} = {\int{\frac{p( {t - {\delta_{nm}( x^{\prime} )}} )}{{{x^{\prime} - x_{m}}}{{x^{\prime} - x_{n}}}}{\eta(x)}d^{3}{x^{\prime}.}}}} & {{Eq}.\mspace{11mu}(10)}\end{matrix}$

Equation (10) is a representation of one temporal signal in the completemap of 10,240×10,240 element-to-element responses. The strength of thereflector η(x) at location x in subtriangle A is estimated by formingthe sum

$\begin{matrix}{{\varphi( {x,t} )} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{\sum\limits_{n \in A}{{{x^{\prime} - x_{m}}}{{x^{\prime} - x_{n}}}{\varphi_{mn}( {t + {\delta_{n\; m}(x)}} )}}}}} & {{Eq}.\mspace{11mu}(11)}\end{matrix}$of received signals with time offsets that cancel the delays to and fromthe reflector. Substituting Eq. (10) into Eq. (11) and assuming that thenorms in the numerator of Eq. (11) are canceled by the norms in thedenominator of Eq. (10) gives

$\begin{matrix}{{\varphi( {x,t} )} = {\int{\lbrack {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{\sum\limits_{n \in A}{p( {t + {\delta_{n\; m}(x)} - {\delta_{n\; m}( x^{\prime} )}} )}}} \rbrack{\eta( x^{\prime} )}d^{3}{x^{\prime}.}}}} & {{Eq}.\mspace{11mu}(12)}\end{matrix}$

The δ_(nm) (x) geometric time shifts that are used to form the sum inEq. (11) perform the same focusing function as the β-factors that areused in Eq. (6) to focus monochromatic responses. Evaluation of Eq. (12)at t=0 gives the expression

$\begin{matrix}{{\varphi( {x,0} )} = {\int{\lbrack {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{\sum\limits_{n \in A}{p( {{\delta_{n\; m}(x)} - {\delta_{n\; m}( x^{\prime} )}} )}}} \rbrack{\eta( x^{\prime} )}d^{3}x^{\prime}}}} & {{Eq}.\mspace{11mu}(13)}\end{matrix}$that represents ψ(x, 0) as the imaged value of η(x) that results fromthe imaging kernel

$\begin{matrix}{{K( {x,x^{\prime}} )} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{\sum\limits_{n \in A}{{p( {{\delta_{n\; m}(x)} - {\delta_{n\; m}( x^{\prime} )}} )}.}}}} & {{Eq}.\mspace{11mu}(14)}\end{matrix}$However, the limited range of transmit and receive directions that areincluded in Eq. (14) yields an oscillatory imaging kernel that does nothave low spatial-frequency components. To eliminate oscillations in theimaged values that are introduced by this kernel the medium variationsare estimated by the value of the complex envelope of ψ(x, t) at t=0rather than by the value of ψ(x, t) at t=0. Thus,{tilde over (η)}(x)=Env[φ(x,t)]_(t=0)  Eq. (15)Envelope detection entails demodulation, i.e., multiplication by e^(jω)⁰ ^(t), where ω₀ is the nominal center or carrier frequency of the pulsep (t), followed by a lowpass filter. However, since only one value ofthe envelope, i.e., the value at t=0, is required, both operations canbe consolidated into the inner product{tilde over (η)}(x)=Env[φ(x,t)]_(t=0)=∫ω(x,t)e ^(jω) ⁰ ^(t) h_(LP)(t)dt,  Eq. (16)where h_(LP) (t) denotes the impulse response of the lowpass filter.These computations can also be accomplished using frequency componentsof the received signals rather than time samples. In that case, Eq. (11)and Eq. (16) have the forms

$\begin{matrix}{{{\varphi( {x,\omega} )} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{\sum\limits_{n \in A}{{{x^{\prime} - x_{m}}}{{x^{\prime} - x_{n}}}{\varphi_{mn}(\omega)}e^{{- j}\;\omega\;{\delta_{n\; m}{(x)}}}}}}}\mspace{20mu}{and}} & {{Eq}.\mspace{11mu}(17)} \\{\mspace{85mu}{{{\overset{\sim}{\eta}(x)} = {\int{{\varphi( {x,\omega} )}{H_{LP}( {\omega - \omega_{0}} )}{dt}}}},}} & {{Eq}.\mspace{11mu}(18)}\end{matrix}$where H_(LP) (ω) is the frequency response or transfer function of thelowpass filter.

Unfortunately, envelope detection is a nonlinear process that producesimaged values that do not map to mechanical properties of the medium.The estimate {tilde over (η)}(x) is, therefore, only used to identifythe structure of the scattering object. Explicit values of the mediumvariations that are assigned to this structure are determined by othermeans.

The computations that assign values of sound speed and attenuation tothe medium variations in subtriangle A make use of aberration estimatesat a series locations x₁ ^(A), x₂ ^(A), . . . along the line from thecenter of the subtriangle to the center of the hemisphere (See FIG. 10). These aberration estimates are also used to compensate the receivesignals in Eq. (11) to improve the accuracy of the estimate 4(x) givenby Eq. (15). Estimation of aberration is, therefore, a key step in thisimaging procedure, and is described below.

Aberration Estimation

If the transmission paths to and from the reflector at x are nothomogeneous, then the time delays of arrival times for reflections fromx will have the formδ_(nm)(x)+ε_(m)(x)+ε_(n)(x)  Eq. (19)where ε_(m) (x) and ε_(n) (x) are adjustments to the geometric delayδ_(nm) (x) that account for changes in sound speed between thetransmitter at x_(m) and x, and between x and the receiver at x_(n).These terms cause misalignment of the time signals in the summation inEq. (11) or, equivalently, phase variations in the frequency componentsthat appear in the summation in Eq. (17). This results in a blurring ofthe estimate 4(x). However, if the terms ε_(m) (x) and ε_(n) (x) canestimated, then the estimated delays can be deducted from the geometricdelay factors to restore signal alignment.

In practice, only the difference in the time delays associated withneighboring elements can be readily estimated. To describe thistime-difference computation, let x_(n) and x_(n)′ be adjacent receiveelements. Also, let

$\begin{matrix}{{{\varphi^{(n)}( {x,t} )} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{{{x - x_{m}}}{{x - x_{n}}}{\varphi_{mn}( {t + {\delta_{n\; m}(x)}} )}}}}\mspace{20mu}{and}} & {{Eq}.\mspace{11mu}( {20a} )} \\{{\varphi^{(n^{\prime})}( {x,t} )} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{{{x - x_{m}}}{{x - x_{n}}}{\varphi_{{mn}^{\prime}}( {t + {\delta_{n\; m^{\prime}}(x)}} )}}}} & {{{Eq}.\mspace{11mu}( {20b} )}\mspace{11mu}}\end{matrix}$be sums of the delayed signals for all 256 transmit elements and one orthe other of the receive elements n and n′. These expressions have thesame form as Eq. (11) except that only one receive element is includedin the sum. A relationship between these two summed signals is derivedby using Eq. (10) to represent each of the element-to-element signals,but with the adjusted delays δ_(nm) (x′)+ε_(m) (x′)+ε_(n) (x′) in placeof the simple geometric delays δ_(nm) (x′). This gives

$\begin{matrix}{{{\varphi^{(n)}( {x,t} )} = {\int{\lbrack {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{p( {t + {\delta_{n\; m}(x)} + {ɛ_{m}(x)} + {ɛ_{n}(x)} - {\delta_{n\; m}( x^{\prime} )}} )}} \rbrack{\eta( x^{\prime} )}d^{3}x^{\prime}}}}\mspace{20mu}{and}} & {{Eq}.\mspace{11mu}( {21a} )} \\{{\varphi^{(n^{\prime})}( {x,t} )} = {\int{\quad{\lbrack {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{p( {t + {\delta_{{n\;}^{\prime}m}(x)} + {ɛ_{m}(x)} + {ɛ_{n^{\prime}}(x)} - {\delta_{n^{\prime}\; m}( x^{\prime} )}} )}} \rbrack{\eta( x^{\prime} )}d^{3}{x^{\prime}.}}}}} & {{{Eq}.\mspace{11mu}( {21b} )}\mspace{11mu}}\end{matrix}$

Since the receiver elements n and n′ are adjacent to each other, thegeometric delays δ_(nm) (x)−δ_(nm) (x′) and δ_(n′m)(x)−δ_(n′m) (x′) willbe nearly identical. If, in addition, the delays ε_(n) (x) and ε_(n′)(x) are essentially constant near the focal point, thenφ^((n′))(x,t)=φ^((n))(x,t+ε _(n′)(x)−ε_(n)(x)).  Eq. (22)Thus, the two summed signals in Eq. (21a) and Eq. (21b) only differ bythe time shift ε_(n′) (x)−ε_(n) (x). This time shift is usuallyestimated by finding the peak of the cross correlation of the twosignals. Frequency-domain computations may also be used. In that case,the summed frequency responses

$\begin{matrix}{{\varphi^{(n)}( {x,\omega} )} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{{{x^{\prime} - x_{m}}}{{x^{\prime} - x_{n}}}{\varphi_{mn}(\omega)}e^{{- j}\;\omega\;{\delta_{n\; m}{(x)}}}}}} & {{Eq}.\mspace{11mu}( {23a} )} \\{{{And}\mspace{20mu}{\varphi^{(n^{\prime})}( {x,\omega} )}} = {\sum\limits_{m \in {A\bigcup A^{1}\bigcup A^{2}\bigcup A^{3}}}{{{x^{\prime} - x_{m}}}{{x^{\prime} - x_{n^{\prime}}}}{\varphi_{{mn}^{\prime}}(\omega)}e^{{- j}\;\omega\;{\delta_{n^{\prime}\; m}{(x)}}}}}} & {{Eq}.\mspace{11mu}( {23b} )}\end{matrix}$are formed instead of the summed signals in Eq. (20a) and Eq. (20b). Thetime shift difference ε_(n)′(x)−ε_(n) (x) is then estimated from therelationshipφ(n′)(x,ω)=φ^((n))(x,ω)e ^(jω[ε) ^(n′) ^((x)−ε) ^(n) ^((x)].)  Eq. (24)

Estimates of the time shift differences for every pair of adjacentreceive elements in triangular subdivisions can be summed along pathsfrom the central element to form estimates {tilde over (ε)}_(n) (x) ofthe time shifts at each of the 64 receiving elements. These estimatescan then be used to adjust the geometrically-determined phase shifts tocompensate for aberration along the ray paths to the receiving elements.Range-dependent compensation for attenuation can be included by usingamplitude factors along with the phase factors for time shiftcompensation on receive.

Aberration estimates for the paths to the transmit elements may beobtained by a similar procedure. Differences in the transmit delays areestimated by forming the summed frequency responses

$\begin{matrix}{{\varphi^{(m)}( {x,\omega} )} = {\sum\limits_{n \in A}{{{x^{\prime} - x_{m}}}{{x^{\prime} - x_{n}}}{\varphi_{mn}(\omega)}e^{{- j}\;{\omega\lbrack\;{{\delta_{n\; m}{(x)}} + {ɛ_{n}{(x)}}}\rbrack}}\mspace{14mu}{and}}}} & {{Eq}.\mspace{11mu}( {25a} )} \\{{\varphi^{(m^{\prime})}( {x,\omega} )} = {\sum\limits_{n \in A}{{{x^{\prime} - x_{m^{\prime}}}}{{x^{\prime} - x_{n}}}{\varphi_{m^{\prime}n}(\omega)}e^{{- j}\;{\omega\lbrack\;{{\delta_{n\; m^{\prime}}{(x)}} + {ɛ_{n}{(x)}}}\rbrack}}}}} & {{Eq}.\mspace{11mu}( {25b} )}\end{matrix}$for each adjacent pair m, m′ of transmit elements. These equationsparallel Eq. (23a) and Eq. (23b), but include time shifts to correct foraberration along the paths to the receive elements. The time shiftε_(n′) (x)−ε_(n) (x) is estimated from the relationshipφ^((m′))(x,ω)=φ^((m))(x,ω)e ^(jω[ε) ^(m) ^(′(x)−ε) ^(m) ^((x)])  Eq.(26)

Similar estimates for time shift differences between every pair ofadjacent transmit elements in the triangle A and the three surroundingtriangular subdivisions are summed along paths from the central elementto form estimates {tilde over (ε)}_(m) (x) for the time shifts at eachof the 256 transmitting elements.

The aberration time shifts {tilde over (ε)}_(n) (x) and {tilde over(ε)}_(m) (x) found from these computations are used to form thecompensated time delays{tilde over (δ)}_(nm)(y)=δ_(nm)(y)+{tilde over (ε)}_(m)(x)+{tilde over(ε)}_(n)(x)  Eq. (27)for imaging via Eq. (17) and Eq. (18) at locations y that are close tox.

The root-mean-square size of the aberration delays generally increaseswith depth. Also, estimates of the time delay differences betweenadjacent elements become inaccurate when the aberration is strong.Additional processing is then needed to detect and correct forcirculation errors in the time delay differences or to form intermediateestimates of delay differences between adjacent clusters of elements.These complications are avoided by evolving the aberration delays alongthe line from the center of the triangle of receive elements to theorigin of the hemisphere. Referring now to FIG. 10 , initial aberrationestimates at x₁ ^(A) are formed at a depth that is shallow enough toinsure that the aberration is weak. The aberration at x₂ ^(A) will bestronger. However, the aberration at x₂ ^(A) can be computed usinggeometric delays that are already compensated by the estimatedaberration at x₁ ^(A). Thus, at each step, only the incrementalaberration is estimated. A focus of this approach is to implementaberration correction as a straightforward computation that does notrequire conditional logic.

Image Consolidation

Image values are computed as magnitudes of complex envelopes given byEq. (15). These computations use compensated geometric delays given byEq. (27) with aberration adjustments that are most appropriate for theimaged location, i.e., adjustments that are estimated from the focusx_(k) ^(A) that is closest to the image point. However, the image valuesare not evaluated along scan lines as is usually the case. Instead, theimage values are computed at the points where the vertices of a fixedthree-dimensional grid intersect the tetrahedron formed by the triangleof receive elements and the center of the hemisphere as shown in FIG. 13. These image values are reported to the head compute node thatconsolidates the estimated medium variations from all 160 tetrahedralsubvolumes into a single three-dimensional array. This arrangementpermits rapid reconstruction of a breast volume that can be rendered bystandard three-dimensional rendering software.

Segmentation and Assignment of Tissue Properties

The medium variations that are produced by the procedure described aboveconsist of intensities that are similar to the pixel intensities ofb-scan images. To obtain an initial approximation of the scatteringobject these imaged intensities must be translated into the tissuemechanical properties of sound speed and attenuation. Sound speedassignments are more critical because variations in sound speeddetermine the phase distortion of the fields {tilde over (ψ)}_(n)(x) and{tilde over (ψ)}_(m)(x).

The first step is to segment the intensity variations into regions withsimilar amplitude and speckle characteristics. The methods used for thissegmentation can be tailored to the specific characteristics of theimaged intensities, as would be understood by a person skilled in theart. Further, the methods can be based on methods developed forsegmenting MRI images of breast tissue that are currently known in theart. For example, a comparison of temporal and spectral scatteringmethods using acoustically large breast models derived from magneticresonance images is provided in Hesford et al., 2014, J. Acoust. Soc.Am., 136(2). The segmentation may yield a decomposition of the mediuminto a small number (e.g., 10 to 20) of disjoint regions

,

, . . . . These regions are assumed to represent tissues that have theunknown sound speeds c₁, c₂, . . . .

To estimate the unknown sound speeds, let

_(n) (x_(k) ^(A))=∥x_(k) ^(A)−x_(n)∥, denote the length of the linesegment from transducer element n to x_(k) ^(A), where x_(k) ^(A) is oneof the aberration focal points along the ray from the center of triangleA to the center of the hemisphere. Also, let

_(n) ^((i)) (x_(k) ^(A)) denote the length of the portion of this linesegment that belongs to

_(i). Then, geometric optics approximations give the linear equation

$\begin{matrix}{{\sum\limits_{{i = 1},2,\ldots}{{\ell_{n}^{(i)}( x_{k}^{A} )}c_{i}^{- 1}}} = {{{\ell_{n}^{(i)}( x_{k}^{A} )}c_{0}^{- 1}} + {{\overset{˜}{ɛ}}_{n}( x_{k}^{A} )}}} & {{Eq}.\mspace{14mu}(28)}\end{matrix}$in the unknown variables c₁ ⁻¹, c₂ ⁻¹, . . . . Application of Eq. (28)to each time shift estimate ε_(n)(x_(k) ^(A)) results in anoverdetermined system of linear equations in the unknown variables c₁⁻¹, c₂ ⁻¹, . . . that has a unique least-square-error solution.Attenuation factors for the initial approximation are much lessimportant and may be derived from tissue type identifications determinedfrom the sound speeds.

The methods described above differ from conventional imaging in severalrespects. For example, b-scan images are usually formed by syntheticfocusing of the received signals along a scan line (or at least asegment of a scan line) that is aligned with one axis of the image. Thisuse of an extended focus reduces the amount of computation at theexpense of lateral resolution. The resolution realized by the HABISalgorithm is better because separate transmit and receive focuses areformed at each voxel. Efficiency is realized in the HABIS algorithm byimaging tetrahedral subvolumes in parallel. They also differ in thescheme for using progressive compensations that are based on estimatesfrom focal points of increasing depth. They further differ in thespecialized geometry of the estimation procedure that uses one centralsubtriangle of receive elements and four subtriangles of transmitelements that include and surround the subtriangle of receive elements.They also differ in the partitioning of the reconstruction intotetrahedral subvolumes, and they further differ in the procedures usedto segment and assign tissue mechanical properties to the imageintensities.

Wide-Angle and Perturbation Formulations of Split-Step Computations

Propagation of monochromatic fields through the approximate scatteringobject is a time consuming computation that is required by thereconstruction procedure. Multipole or k-space methods may be used toobtain these fields, but these algorithms are too computationally costlyto propagate fields from all 10,240 transducer elements. The split-stepalgorithm is able to perform these propagations more efficiently.Split-step methods separate monochromatic waves into components thattravel in forward and backward directions with respect to adistinguished z axis. If the backward traveling components are ignored,then the propagation can proceed in steps that derive field values onsuccessive x-y planes from the field values on preceding planes.However, exact forward propagators for these steps are nonlinearcombinations of spatial and frequency domain operators that do notcommute. For efficient computation, these propagators must beapproximated by alternating sequences of operators that act in thespatial domain and the spatial-frequency domain. In the simplest cases,the propagators are approximated as a product of a single spatial-domainoperator and a single frequency-domain operator. However, thesepropagators are only accurate when the propagated fields are limited tospatial frequencies with small transverse components. Betterapproximations that include additional terms are often called wide-angleapproximations because they are applicable to fields that have spatialfrequencies with larger transverse components.

Two adaptations for improving the accuracy of split-step computationsare described herein:

1. Wide-angle correction: Exact forward propagators are exponentialoperators of the form exp[jk₀Δz(√{square root over (1+P+Q)}−1)], where Qis a multiplication operator in spatial coordinates and P is amultiplication operator in spatial-frequency coordinates. If the P and Qoperators only produce small variations, then a first-order binomialexpansion of the radical can be used to obtain the approximatepropagator exp [jk₀Δz(P+Q)/2]. Further approximation of the exponentialgives either exp(jk₀ΔzP/2) exp (jk₀ΔzQ/2) or exp(jk₀ΔzQ/2) exp(jk₀ΔzP/2). These factored propagators are used in conventionalsplit-step algorithms and, as noted above, are only applicable to fieldscomprised of spatial frequencies with small transverse components.Wide-angle correction refers to the inclusion of additional terms thatextend the accuracy of the approximations to spatial frequencies withlarger transverse components.

The increased accuracy realized by any correction term must be weighedagainst the added computational cost of including the correction. Thesecosts can be conveniently quantified by counting the number of Fouriertransforms in the computation because the other operations are usuallymuch less expensive than the Fourier transforms. The cost of simple,i.e., narrow-angle, split-step propagation is a single pair of Fouriertransforms, and wide-angle correction terms generally require at leastone additional pair of transforms. Thus, propagations that employwide-angle correction require at least twice as much computation as thenarrow-angle algorithm. The high computational cost of wide-anglecorrection implies that the benefit from the correction should beoptimized. A term has been developed that is more accurate than theknown published correction terms of comparable cost. The term is givenby:

$ {\phi( {x,0} )}arrow{{\phi( {x,0} )} + {jk_{0}\Delta{z\begin{pmatrix}{1/2} \\2\end{pmatrix}}^{- 1}( {{\frac{\sqrt{1 + P} - \frac{P}{2} - 1}{P}\frac{\sqrt{1 + Q} - \frac{Q}{2} - 1}{Q}} + {\frac{\sqrt{1 + Q} - \frac{Q}{2} - 1}{Q}\frac{\sqrt{1 + P} - \frac{P}{2} - 1}{P}}} ){{\phi( {x,0} )}.}}} $

2. Split-step perturbations: In addition to wide-angle correction, aperturbation form of the split-step algorithm can be derived that givessubstantially more accuracy than simple split-step propagation withoutrequiring any additional FFTs. Although the method may not be asaccurate as propagations that include the wide-angle correction, muchbetter results than narrow-angle propagators can be obtained withessentially no additional cost.

Scattering Measurements

Another computation that the HABIS reconstruction procedure requires isthe evaluation of the scattering measurements {tilde over (M)}_(mn) thatare due to the approximate scattering object. These measurements arededucted from the actual scattering measurements M_(mn) to obtain theresidual measurements M_(mn)−{tilde over (M)}_(mn) that are used todetermine the refinement Δη(x). The scattering measurement {tilde over(M)}_(mn) is expressed by the volume integral{tilde over (M)} _(mn)=∫η(x){tilde over (ψ)}_(m)(x)ψ_(n)(x)d ³ x,  Eq.(29)where {tilde over (ψ)}_(m) (x) is the transmit field from transducerelement m that propagates through the approximate scattering object,ψ_(n)(x) is the receive sensitivity pattern of transducer element n thatpropagates homogeneously, and η(x) denotes the medium variations of theapproximate scattering object. Evaluation of the volume integral in Eq.(29) is costly because the integration has to be repeated for all10,240×10,240 combinations of transmit and receive elements.

A much more efficient method for evaluating Eq. (29) has been developed.The method is based on: 1) Expression of the integral in thespatial-frequency domain; 2) Expansion of the Fourier transform ofψ_(n)(x) as an angular spectrum to reduce the three-dimensionalspatial-frequency integral to a surface integral over a sphere ofspatial frequencies; 3) Interpolation of spatial-frequency components onthe sphere to values on hemispheres that are parameterized bytwo-dimensional grids with the same planar orientations as the facets;4) Evaluation of scattering measurements on two-dimensional spatialgrids that each contain one of the array facets by taking the inversetwo-dimensional Fourier transforms of the spatial-frequency arraysobtained in Step 3; and 5) Interpolation of the values on regular arraysof scattering measurements to obtain the values of scatteringmeasurements at the locations of the pseudo-random distribution ofelement positions.

The improvement in efficiency over direct evaluation of Eq. (29) isseveral orders of magnitude. However, very accurate methods ofinterpolation are required.

Fast Rotation Algorithm

To incorporate the approximate scattering object in a split-steppropagation of the field of a transducer element, the array of mediumvariations that prescribe the scattering object must be rotated so thatone dimension of the array is aligned with the element axis. Thus, inthe course of a complete reconstruction, the medium variations must berotated to 40 different orientations that are determined by the 40facets of the hemispheric array (because the facet normal is thepropagation axis of every element in a facet). The split-steppropagations produce three-dimensional arrays of field values at thesame grid locations as the rotated medium variations, and these arraysmust be further rotated to a single common orientation to facilitatesubsequent computations. Since the dimensions of all thesethree-dimensional arrays are large (˜1,500×1,500×1,500) the rotationsare costly. In fact, the rotations are generally more time consumingthan all the other computations. For this reason, a very fastthree-dimensional rotation algorithm was developed that is tailored toGPU computation.

GPU algorithms are most efficient when memory accesses are coalesced andare transferred to and from global memory as infrequently as possible.To minimize the cost of global memory transfer, the algorithm can factorthree-dimensional rotations into a series of skew operations thatinterpolate shifted values along one of the three dimensions of thearray. Such a rotation algorithm can be several times faster thanconventional algorithms rotations, and can also provide increasedprecision more efficiently than conventional algorithms. Furtherdescription of such rotation algorithms as contemplated herein areprovided in Appendix A.

Fast and Accurate Harmonic Interpolation

The algorithm for computing scattering measurements and the algorithmfor rotating three-dimensional arrays described above both requireinterpolation of a set of data values at the vertices of a regularlysampled grid. A large number of interpolated values are generallyrequired. However, in some cases the interpolated locations are spacedirregularly, while in other cases they occur at regular intervals, e.g.,at the midpoints of the sample intervals in the original array.

Since each interpolated value is computed as a linear combination ofsample values in the original array, the computational cost of eachinterpolation is roughly proportional to the number of original samplevalues that take part in the interpolation. For example, atwo-dimensional interpolation that employs values in a 4×4 subarray isfour times as expensive as a two-dimensional interpolation that onlyemploys values in a 2×2 subarray. Thus, to improve efficiency the sizeof the subarray of original values that contribute to each interpolationshould be minimized. On the other hand, interpolations that employsmaller subarrays of original values are less accurate thaninterpolations that employ larger subarrays of original.

In one embodiment, the accuracy of interpolations that are obtained fromsmall subarrays can be increased by optimizing the interpolationcoefficients for a revised array of sample values that are obtained by aprefiltering operation. The improvement in precision realized by thistechnique is significant. Further, in the scattering measurementcomputations, the prefiltering step can be incorporated in priorcomputations at a very low cost. In another embodiment, the accuracy offinite-length interpolations can be increased by optimizing theinterpolation coefficients with respect to a non-standard targetfrequency response. This scheme is ideally suited to the skew operationsof the rotation algorithm. Further description of such interpolations ascontemplated herein is provided in Appendix B.

Each of the algorithmic improvements described herein contribute to theperformance of the HABIS reconstruction procedure. However, theseimprovements also all have applications in other areas. For example,split-step propagations can be used to model waves of all forms, e.g.,electromagnetic, seismic, ultrasonic, etc. Similarly, the efficient formof the scattering computation can also be applied to such cases. Theinterpolation and volume rotation algorithms also have wide rangingapplications that are completely independent of the reconstructionalgorithm, as would be understood by a person skilled in the art.

Noise Reduction

Image reconstruction by HABIS is based on the scattering measurementrepresentations given by Eq. (2) and Eq. (10). Any contributions to thescattering measurements that are not included in these representationscan degrade image resolution, and may be considered noise. Thesecontributions include conventional forms of noise such as thermal noise,crosstalk, and jitter that originate in the electronics as well asinclude other terms that are correctly measured responses to effects forwhich Eq. (2) and Eq. (10) do not account. Examples of this latter formof noise are patient movement and inaccuracy in the split-steppropagations of the transmitted fields. An important aspect of the HABISdesign is anticipating sources of noise and finding ways to minimize theimage-degrading effects of these terms. Reconstruction of high-qualityimages can be difficult or impossible without the noise reductioncompensations described below.

Transmission Encoding

Thermal noise from receive circuitry is inescapable. However, the effectthat this noise has on image reconstruction can be diminished if theamplitudes of the received signals can be increased without increasingthe noise. One way to do this is to increase the amplitude of thetransmitted signals. However, practical considerations limit drivingsmall transducer elements with high-voltage signals. Furthermore,high-amplitude ultrasound waves propagate nonlinearly and produceundesirable nonlinear responses. In conventional pulse-echo ultrasoundimaging systems, reflections are amplified by transmitting ultrasoundwaves simultaneously from many elements that are usually concentrated byfocusing in regions of interest. However, this yields a smaller set ofscattering measurements that have less coherent resolution than thecomplete set of 10,240×10,240 element-to-element responses in thedescribed imaging system. The signal-to-noise ratios of HABIS scatteringmeasurements are improved by a different method. A total of 10,240transmissions are used in each scan. Each of these transmissions is acoded combinations of signals from 2,048 transmit elements. The samebasic pulse shape is transmitted from all of the 2,048 elements thatparticipate in a given transmission. The sign of the pulse is invertedfor a subset of the elements. The subset of negated pulses changes fromtransmission to transmission according to a scheme that allows thereceived signals to be added and subtracted in combinations that isolatethe responses to transmissions from each of the individual elements. Thepurpose of this encoding and decoding of the element-to-elementresponses is to increase the signal-to-noise ratio of the receivedsignals. Special system characteristics have to be satisfied for thisencoding to be effective because the encoding is highly sensitive tocrosstalk and jitter between samples of received signals from differentreceive elements. Meeting the reduced tolerances for jitter andcrosstalk required specially designed electronic circuitry.

Two-Second Illumination

Imaging systems that use transducer probes or small two-dimensionalarrays of transducer elements that must be translated or rotated or bothto acquire scattering from different orientations have relatively longscan times that introduce measurement error caused by tissue movement.To minimize error from tissue movement, the HABIS scanning procedure wasdesigned to collect a complete scan of 10,240 receive signals from eachof 10,240 transmissions within a time span of about two seconds. This isessentially the smallest possible scan time for this set oftransmissions because two seconds is about equal to the time requiredfor 10,240 ultrasound transmissions to make round trip excursionsthrough an average size breast. Numerous design challenges were met toaccomplish this rapid acquisition. Signals from all 10,240 receivers hadto be acquired simultaneously, so the receive channels could not bemultiplexed. Thus, a separate data-acquisition channel was needed foreach receiver. Furthermore, since each complete scan is comprised ofnearly a terabyte (TB) of data, the system must include local memoriesto retain the acquired data as well as a high-speed data transfer systemto transmit the acquired data to the high-performance computer networkin a timely manner. Currently available ultrasound imaging systemsemploy far fewer channels and do not handle data sets of this size.

Time Gates that Eliminate Noise from Remote Scattering

All ultrasound imaging systems reconstruct medium variations by dividingthe reflected signals into contributions that come from differentregions. The HABIS inverse scattering algorithm is a coherent imagingmethod that resolves locations in the scattering object by forminglinear combinations of residual scattering measurements that coherentlysum the contributions that come from one location while incoherentlysumming the contributions that come from all other locations. Thisimaging technique has higher point resolution than conventionalalgorithms, but can have reduced contrast resolution when the incoherentsums are appreciable. Thus, incoherently summed scattering from distantreflectors is a form of noise that can degrade image quality. To reducethis degradation, receive time gates are used, prior to reconstruction,to isolate the scattering measurements that originate from differentregions. The noise reduction strategy used in HABIS is not readilyapplicable to other inverse scattering algorithms that performreconstructions globally. However, time gates can be convenientlyincluded in the HABIS reconstruction algorithm because HABISreconstructions are partitioned into a large number of subvolumes.

Calibration that Eliminates Variations in System Responses

Representation of the medium variations by Eq. (2) can only be asaccurate as the estimates {tilde over (ψ)}_(m) (x) and {tilde over(ψ)}_(n) (x) of transmit fields and receiver sensitivity patterns thatappear in the integrand of Eq. (2). These fields are computed by usingthe split-step algorithm to propagate fields from the element locationsx_(m) and x_(n) through the approximate scattering object. Thus, anyerror in an element location can translate into phase error throughoutthe entire field. For this reason, accurate determination of the elementpositions is needed. Since minor variations in manufacturing canintroduce small offsets in effective element positions, a calibrationscheme was developed to determine the element positions from scans of aspecially constructed calibration fixture. These scans are also used toequalize characteristics of all the elements in the hemispheric array.This equalization compensates for variations in element sensitivity andalso element-to-element variation in the electro-acoustic frequencyresponses.

Use of Matching Fluids

The accuracy of the field estimates {tilde over (ψ)}_(m) (x) and {tildeover (ψ)}_(n) (x) also depend on the accuracy of the split-stepalgorithm. Split-step propagations lose precision when the mediumvariations cause strong refraction of propagated fields, and theaccumulated error that results from this refraction can be especiallylarge when the refraction occurs in the early stages of thetransmission. The interface between breast tissue and the ambient fluidis always encountered at the beginning of each transmission, and causesstrong refraction when the ambient fluid is water because of thedifference between the speed of sound in water and the speed of sound intissue. For this reason, an ambient fluid other than water is desirableto minimize refraction at the breast surface.

Estimation of Scattering Object Detail in the Antipodal Hemisphere

The hemispheric array breast imaging system (HABIS) includes featuresthat provide advantages over currently available ultrasound imagingsystems, including b-scan instruments and systems that employ otherforms of inverse scattering. One feature is the measurement ofscattering throughout a hemisphere surrounded by an array of ultrasoundtransducers. In one embodiment, another feature is the use of scatteringmeasured in the hemisphere surrounded by the array to estimatescattering object detail that would come from scattering in theantipodal hemisphere, where measurements of scattering cannot be madedirectly.

As described herein, the HABIS reconstruction algorithm uses scatteringmeasured throughout an entire hemisphere surrounded by the transducerarray to form an estimate of the scattering object. Additionally, theHABIS reconstruction algorithm extrapolates from the measurementhemisphere spatial frequencies that correspond to scattering in theantipodal hemisphere to obtain information equivalent to that obtainedfrom measurement of scattering in that hemisphere. In other words, theHABIS algorithm uses scattering measured in a solid angle of 2πsteradians (i.e., throughout a hemisphere encompassed by the transducerarray), and uses spectral extrapolation to obtain information fromscattering that cannot be measured in the antipodal hemisphere directly(i.e., throughout the antipodal solid angle of 2 π steradianscorresponding to the other hemisphere). The result is that spatialfrequencies spanning an entire lowpass volume in the Fourier transformof the image space, a transform space known as wave space, are used bythe HABIS image reconstruction algorithm to obtain the maximumtheoretical amount of spatial detail contained in that volume.

Accordingly, the use of all the scattering object detail in a lowpassvolume of the scattering object transform space in the HABISreconstruction algorithm provides an advantage over other ultrasoundimaging methods that only measure scattering in a plane (and so ignoreout-of-plane scattering), or that only measure scattering over arelatively narrow solid angle in the forward direction (and so ignorescattering in most of the total 4π-steradian solid angle). These othermethods do not obtain or use all of the measurable scattering objectinformation. Therefore, the other methods are incapable of imagingdetail associated with scattering throughout an entire solid angle of 4πsteradians.

Another advantageous feature of HABIS is the short data-acquisitiontime. For example, in one embodiment, the data-acquisition time is abouttwo seconds. This data-acquisition time, which is very much shorter thanin systems that rotate or otherwise move transducers to acquire data,practically eliminates motion-induced degradation of the reconstructedvolume. Yet another advantage of HABIS is point resolution that is bothhigh and isotropic. The point resolution of HABIS is essentially equalto the lateral resolution in high-quality x-ray mammograms in whichimage features are typically blurred by summation of x-ray absorptionalong the direction of the projection.

Illustration of HABIS Imaging Capabilities

Reconstructions based on inverse scattering and corresponding b-scanimages have been obtained using calculated scattering by a 10-pointresolution target, and by a realistic breast model derived from 200-amresolution MRI data to illustrate the resolution and fidelity of thevolumetric images produced by HABIS. Each of the inverse-scatteringreconstructions used scattering at the surface of the entire hemispherictransducer array, and also used an estimation of scattering objectdetail that would come from measurements in the opposite hemispherewhere scattering cannot be directly measured. This approach obtainsspatial frequencies spanning an entire lowpass volume in the Fouriertransform of the image space (a transform space known as wave space) torealize the maximum theoretical amount of spatial detail contained inthat volume. Each of the b-scans used transmission from a 64-elementsubtriangle and three 64-element subtriangles surrounding that centralsubtriangle, and used reception from the central 64 element subtriangle.Use of these transmit and receive apertures takes advantage of thesystem architecture to allow formation of a volumetric b-scan in aboutone minute after a two-second interval of data collection. This b-scanhas sufficient resolution to assess the quality of the acquired data fora more time-consuming volumetric reconstruction based on inversescattering.

Representative images of the resolution target are shown in FIG. 14 . Atthe 5-MHz frequency used to obtain FIG. 14A, the Rayleigh-definedresolution inferred from inverse-scattering reconstructed data is 100μm. For the location of the resolution target in the b-scan shown inFIG. 14B, the transmit f-number was about 1.5 and the receive f-numberwas about 2.9. The pulse spectrum used for the b-scan had araised-cosine shape centered around 5 MHz with a −6-dB bandwidth of 2MHz. These transmit-receive apertures and pulse bandwidth yield anisotropic Rayleigh-defined resolution of about 700 μm. A b-scan usingtransmission and reception throughout the hemisphere would have alateral resolution comparable to the resolution in the reconstruction asa result of the transmit and receive effective f-numbers each being onehalf.

Representative sections of a 6.4-mm3 subvolume in the breast model areshown in FIG. 15 . The reconstruction was performed at a frequency of 5MHz as in the case of the resolution target. The detail evident in thereconstructed-image panel of FIG. 15 comes from the use of scatteringthroughout the entire surface of the hemispheric array. This scatteringwas first used to estimate scattering measurements that would be made bya virtual array in the opposite hemisphere and then the combined set ofdata was used in the reconstruction. The b-scan used aberrationcorrection. For the location of the subvolume of the model, the transmitf-number in the b-scan was about 3.0 and the receive f-number in theb-scan was about 6.0. These transmit-receive apertures yield aRayleigh-defined lateral resolution of about 1.5 mm at the 5-MHz centerfrequency. The pulse spectrum was the same as used for the resolutiontarget so the axial resolution is about 700 m as in the case of theresolution target. The inverse-scattering reconstruction clearly showsthe subvolume morphology with high fidelity while the b-scan mainlyhighlights discontinuities that are approximately perpendicular to theb-scan lines.

Rotation in Two Dimensions

Let u₁, u₂ be the pair of generating vectors used to step betweenvertices of a Cartesian grid v (n, m). Thus,v(n,m)=nu ₁ +mu ₂ :n,m=0,±1,±2, . . .   (1)The vectors u₁ and u₂ are assumed to be orthogonal, but are not assumedto be of equal length. A rotated grid of vertices{tilde over (v)}(n,m)=nũ ₁ +mũ ₂ :n,m=0,±1,±2, . . .   (2)can be formed by replacing u₁ and u₂ in Eq. (1) with counterclockwiserotations ũ₁=R_(θ)[u₁] and ũ₂=R_(θ)[u₂] of u₁ and u₂ where θ is theangle of rotation.

Let Q[n, m] be an array of sample values of a smooth function q x, y) atthe vertices v(n, m). An efficient interpolation scheme is developedbelow for generating sample values {tilde over (Q)}[n, m] of q(x, y) atthe rotated vertices {tilde over (v)}(n, m). The sample values at thevertices of the rotated grid are obtained from three interpolations thateach generate sample values at the vertices of a new grid that is skewedrelative to the vertices of the previous grid. The skewing operation isillustrated in FIG. 16 . The black dots in the figure are the verticesof the original grid, and the red dots are the vertices of the skewedgrid. Notice that both grids have the same vertical lines. However, thered lines that connect vertices in the non-vertical dimension are nothorizontal. The vertices of the skewed grid in FIG. 16 can berepresented byv′(n,m)=nu′ ₁ +mu′ ₂ :n,m=0,±1,±2, . . . ,  (3)where u′₁=u₁+αu₂ and u′₂=u₂.

Function values at the vertices of the skewed grid can be generated fromfunction values at the vertices of the original grid by one-dimensionalinterpolations along the vertical grid lines. The simplicity andefficiency of these interpolations is the reason that skewing ispreferable to direct rotation. A skew can also be applied to thevertical lines rather than horizontal lines. In that case, values at theskewed vertices are obtained from one-dimensional interpolation alonghorizontal lines in the original grid.

FIG. 17 shows a succession of skew transformations that result in arotated grid*. The black dots in the figure are the vertices of theoriginal grid and the red dots are the vertices of the grid that resultsfrom skewing the horizontal lines of the original grid. A second skewingof the vertical lines in the red grid then results in the grid of greenvertices. Finally, skewing the green vertices in the other (roughlyhorizontal) dimension gives the blue grid that is a rotation of theoriginal black grid. The grid lines associated with the green verticesare not shown because one set of lines in the green grid coincides withlines of the red grid lines, and the other set of lines in the greengrid coincide with lines of the blue grid. *A rotated orthogonal gridcan be obtained with two skew transformations rather than three, but thepitches of the grid are distorted by this transformation.

The successive skew transformation shown in FIG. 17 can be expressed aschanges in the grid generating vectors as follows:

$\begin{matrix}\begin{matrix}{{First}\mspace{14mu}{Skew}\text{:}} & \begin{pmatrix}u_{1} \\u_{2}\end{pmatrix} & \Rightarrow & \begin{pmatrix}{u_{1}^{\prime} = {u_{1} + {\alpha\; u_{2}}}} \\{u_{2}^{\prime} = u_{2}}\end{pmatrix} \\{{Second}\mspace{14mu}{Skew}\text{:}} & \begin{pmatrix}u_{1}^{\prime} \\u_{2}^{\prime}\end{pmatrix} & \Rightarrow & \begin{pmatrix}{u_{1}^{''} = u_{1}^{\prime}} \\{u_{2}^{''} = {u_{2}^{\prime} + {\beta\; u_{1}^{\prime}}}}\end{pmatrix} \\{{Third}\mspace{14mu}{Skew}\text{:}} & \begin{pmatrix}u_{1}^{''} \\u_{2}^{''}\end{pmatrix} & \Rightarrow & \begin{pmatrix}{{\overset{\sim}{u}}_{1} = {u_{1}^{''} + {\gamma\; u_{2}^{''}}}} \\{{\overset{\sim}{u}}_{2} = u_{2}^{''}}\end{pmatrix}\end{matrix} & (4)\end{matrix}$

The geometry used to determine the skews δ₁=αu₂, δ₂=βu′₁, and δ₃=γu″₂ isshown in FIG. 18 . Similar triangles in the figure imply that ũ₁=u₁ andũ₂=u₂, and also imply that ũ₁=R_(θ)[u₁] and ũ₂=R_(θ)[u₂]. The identities∥δ₁∥=∥δ₃ ∥=∥u ₁∥ tan(θ/2),  (5a)∥δ₂∥=2∥u ₂∥ sin(θ/2),  (5b)∥u′ ₁ ∥=∥u ₁∥/cos(θ/2),  (5c)and∥u″ ₂ ∥=∥u ₂∥.  (5d)that are also evident from the figure giveα=∥δ₁ ∥/∥u ₂∥=(∥u ₁ ∥/∥u ₂∥)tan(θ/2),  (6a)β=−∥δ₂ ∥/∥u′ ₁∥=(∥u ₂ ∥/∥u ₁∥)sin(θ),  (6b)andγ=∥δ₃ ∥/∥u″ ₂∥=(∥u ₁ ∥/∥u ₂∥)tan(θ/2).  (6c)In particular, if ∥u₁∥=∥u₂∥, then α=γ=tan(θ/2) and β=−sin(θ). Sincecoordinate rotations are generally applied to isotropic grids, theidentity u₁=u₂ is assumed throughout the subsequent discussion.

A step-by-step the procedure for two-dimensional rotation of the samplevalues Q[n, m] at the vertices of an isotropic grid is given below:

-   -   1. For each fixed value n of the first index, the        one-dimensional array Q[n, m]:m=0, ±1, ±2, . . . is translated        by the fractional offset n tan(θ/2) to obtain the        two-dimensional array Q′[n, m]=Q[n, m+n tan(θ/2)]:n, m=0, ±1,        ±2, . . . .    -   2. For each fixed value m of the second index, the        one-dimensional array Q′[n, m]n=0, ±1, ±2, . . . is translated        by the fractional offset −m tan(θ) to obtain the two-dimensional        array Q″[n, m]=Q′[n−m sin(θ), m]:n, m=0, ±1, ±2, . . . .    -   3. For each fixed value n of the first index, the        one-dimensional array Q″[n, m]:m=0, ±1, ±2, . . . is translated        by the fractional offset n tan(θ/2) to obtain the rotated        two-dimensional array {tilde over (Q)}[n, m]=Q″[n, m+n        tan(θ/2)]:n, m=0, ±1, ±2, . . . .

A procedure for one-dimensional translation must also be specified.Exact band-limited translation can be achieved by means of harmonicinterpolation, but convolution with a shifted and modified sine functioncan give accurate results with less computation. The accuracy of theseinterpolations is determined by the length of the convolution filters.If the translations are performed with convolution filters of length L,then the computational cost of the entire rotation is

$\begin{matrix}{{\underset{\underset{\underset{\#\mspace{14mu}{of}\mspace{14mu}{skews}}{︷}}{︸}}{3} \times \underset{\underset{\underset{{Mult}\mspace{14mu}{and}\mspace{14mu}{adds}}{︷}}{︸}}{L} \times \underset{\underset{\underset{\times {, +}}{︷}}{︸}}{2} \times \underset{\underset{\underset{\#\mspace{14mu}{of}\mspace{14mu}{values}}{︷}}{︸}}{N}} = {6\;{LN}\mspace{14mu}{{flops}.}}} & (7)\end{matrix}$Angle Reduction and Array Dimension Requirements

The procedure described in the previous section factors two-dimensionalcoordinate rotations into three skew transformations. However, thefactorization does not consider possible limitations imposed by thefinite dimensions of the array. The following additional factorizationsare noted before addressing this issue:

-   -   1. If ũ₁=R_(π)[u₁] and ũ₂=R_(π)[u₂], then ũ₁=−u₁, ũ₂=−u₂ and,        consequently, {tilde over (Q)}[n, m]=Q[−n, −m]:n, m=0, ±1, ±2, .        . . .    -   2. If ũ₁=R_(π/2)[u₁] and ũ₂=R_(π/2)[u₂], then ũ₁=−u₂, ũ₂=u₁ and,        consequently, {tilde over (Q)}[n, m]=Q[−m, n]:n, m=0, ±1, ±2, .        . . .        Thus, reflection and transposition can be used to reduce any        rotation to a rotation that is limited to the range |θ|≤π/4.

Let [n₀, m₀] be the indices of a particular value in the original array.A rotation of the coordinates will result in a reassignment of thisvalue to new coordinates [ñ₀, {tilde over (m)}₀], and if thesecoordinates are not contained within the dimensions of the array thenthe value must be discarded. The array dimensions must, therefore, belarge enough to include both the initial and rotated indices of all thevalues to be rotated. This is a natural and obvious requirement that isapplicable to any in place rotation algorithm. In some cases,factorization of the rotation into skew transformations can require alarger array because values can be translated to intermediate indicesthat exceed the array dimensions. The translations of [n₀, m₀] thatresult from the skew transformations can be expressed as

$\begin{matrix}\begin{matrix} \lbrack {n_{0},m_{0}} \rbrackarrow\lbrack {n_{0},{m_{0} + {n_{0}{\tan( {\theta/2} )}}}} \rbrack  \\ arrow\lbrack {{\overset{\sim}{n}}_{0},{m_{0} + {n_{0}{\tan( {\theta/2} )}}}} \rbrack  \\{ arrow\lbrack {{\overset{\sim}{n}}_{0},{\overset{\sim}{m}}_{0}} \rbrack ,}\end{matrix} & (8)\end{matrix}$where [ñ₀, {tilde over (m)}₀] are the final rotated indices. Thisequation implies that successful transfer of the array value withindices [n₀, m₀] to the rotated indices [ñ₀, {tilde over (m)}₀] requiresboth n₀ and ñ₀ to be in the range of the first array index, and alsorequires m₀, {tilde over (m)}₀ and m₀+n₀ tan(θ/2) to be in the range ofthe second array index. These requirements are illustrated in FIG. 19 .The red dotted curve in the figure is the image of the unit circle afterapplication of the first skew transformation for a π/4 rotation. Thegreen dotted curve is the image of the red dotted line after applicationof the second skew operation for the same rotation. All indices insidethe unit circle are mapped to points inside the red dotted curve by thefirst skew transformation, and are then mapped to points inside thegreen dotted curve by the second skew transformation. The red, green andblue arrows in the figure show the succession of all three skewtransformations at a few selected points.

Successful rotation of all points inside the unit circle requires anarray with padding in the second (i.e., vertical) dimension toaccommodate the skew transformations that bridge the top and bottom capsof the circle^(†). The exact scale factor for the array expansion is sec(π/8)=1.0824 at both ends. However, if the initial data values only spanthe upper hemisphere then padding is only needed at the top of thehemisphere. This is also illustrated in the figure by the dotted bluerectangle which outlines the array of data values needed for a π/4rotation of the upper hemisphere. †Additional space is needed in thevertical rather than horizontal direction because the skew operationsare not symmetric. An alternate implementation of the rotation thatexchanges the first and second dimensions in the skew transformationswould require additional space in the horizontal rather than verticaldirection.

The original, red dotted, and green dotted circles could also be used tolimit the ranges of the interpolations in the three skew operations.However, the additional logic needed to impose these limits willdiminish efficiencies realized from reduced computation.

Rotation in Three Dimensions

Let u₁, u₂, and u₃ be the generating vectors used to step betweenvertices of the Cartesian grid v (n, m, o) so thatv(n,m,o)=nu ₁ +mu ₂ +ou ₃ :n,m,o=0,±1,±2, . . .   (9)Assume the vectors u₁, u₂, and u₃ are orthogonal and of equal length.Then, a rotated grid of vertices{tilde over (v)}(n,m,o)=nũ ₁ +mũ ₂ +oũ ₂ :n,m,o=0,±1,±2, . . .   (10)can be formed by replacing u₁, u₂, and u₃ in Eq. (9) with the rotatedvectors ũ₁=R_(ϕ,θ,ψ)[u₁], ũ₂=R_(ϕ,θ,ψ)[u₂], and ũ₃=R_(ϕ,θ,ψ)[u₃], whereϕ, θ, and ψ are Euler angles that uniquely determine the appropriaterotation. Euler rotations factor into the product R_(ϕ,θ,ψ)=R_(ψ) ³R_(θ) ² R_(ϕ) ³ of three two-dimensional rotations where R_(∂) ^(k)denotes a counterclockwise rotation around the k axis through the angle∂. However, in many applications, less general rotations are sufficient.For example, rotations that orient tissue models for split-steppropagation only need to satisfy the conditionũ ₃ =R _(ϕ,θ,ψ) [u ₃ ]=−n,  (11)where n is a unit vector that points in the direction of forwardpropagation. If spherical coordinates^(‡) ϕ and θ specify n as ‡ Theazimuth angle ϕ used in these coordinates is measured relative to the u₂axis rather than the u₁ axis and, consequently, the axis of the polarangle rotation R_(−θ) ¹ is ũ₁ rather than ũ₂.−n=sin θ sin ϕ sin ϕu ₁+sin θ cos ϕu ₂+cos θu ₃,  (12)then Eq. (11) is satisfied by the rotation R_(−θ) ¹R_(ϕ) ³, asillustrated in FIG. 5 . To implement this rotation, the step-by-stepprocedure for two-dimensional rotation given earlier is supplementedwith additional skew transformations as follows:

-   -   1. For each pair of fixed values n, o of the first and third        indices, the one-dimensional array Q[n,m,o]:m=0, ±1, ±2, . . .        is translated by the fractional offset n tan(ϕ/2) to obtain the        three-dimensional array Q′[n, m, o]=Q[n, m+n tan(ϕ/2), o]:n, m,        o=0, ±1, ±2, . . . .    -   2. For each pair of fixed values m and o of the second and third        indices, the one-dimensional array Q′[n, m, o]:n=0, ±1, ±2, . .        . is translated by the fractional offset −m sin(ϕ) to obtain the        two-dimensional array Q″[n, m, o]=Q′[n−m sin(ϕ), m, o]:n, m,        o=0, ±1, ±2, . . . .    -   3. For each pair of fixed values n, o of the first and third        indices, the one-dimensional array Q″[n, m, o]:m=0, ±1, ±2, . .        . is translated by the fractional offset n tan(ϕ/2) to obtain        the rotated three-dimensional array Q′″[n, m, o]=Q″[n, m+n        tan(ϕ/2), o]:n, m, o=0, ±1, ±2, . . . .    -   4. For each pair of fixed values n, o of the first and third        indices, the one-dimensional array Q′″[n, m, o]:m=0, ±1, ±2, . .        . is translated by the fractional offset o tan(θ/2) to obtain        the rotated three-dimensional array Q^(iv)[n, m, o]=Q′″[n, m+o        tan(θ/2), o]:n, m, o=0, ±1, ±2, . . . .    -   5. For each pair of fixed values n, m of the first and second        indices, the one-dimensional array Q^(iv)[n, m, o]:o=0, ±1, ±2,        . . . is translated by the fractional offset −m sin(θ) to obtain        the two-dimensional array Q^(v)[n, m, o]=Q^(iv)[n, m, o−m        sin(θ)]:n, m, o=0, ±1, ±2, . . . .    -   6. For each pair of fixed values n, o of the first and third        indices, the one-dimensional array Q^(v)[n, m, o]:m=0, ±1, ±2, .        . . is translated by the fractional offset o tan(θ/2) to obtain        the rotated three-dimensional array Q^(vi)[n, m, o]=Q^(v)[n, m+o        tan(θ/2), o]:n, m, o=0, ±1, ±2, . . . .

Each of these six steps is a skew transformation that is comprised ofone-dimensional transformations along different axes. However, the skewstransformations in Step 3 and Step 4 are in the same direction and,therefore, con be consolidated into a single step: 3+4 For each pair offixed values n, o of the first and third indices, the one-dimensionalarray Q″[n, m, o]:m=0, ±1, ±2, . . . is translated by the fractionaloffset n tan(ϕ/2)+o tan(θ/2) to obtain the rotated three-dimensionalarray Q^(iv)[n, m, o]=Q″[n, m+n tan(ϕ/2)+o tan(θ/2), o]:n, m, o=0, ±1,±2, . . . .

This consolidation is the reason for measuring azimuth angle relative tothe u₂ axis rather than the u₁ axis, and also for using a polar anglerotation R_(−θ) ¹ with axis u₁.

Interpolation Coefficients

High-performance interpolation coefficients for the skew transformationscan be determined using methods described in Ref. [1]. However, theprefiltering used in Ref. [1] cannot be efficiently interleaved with theskew transformations and is, therefore, not applicable. Frequency-domaindistortion of interpolations for fractional steps ε=0.1, 0.2, 0.3 and0.4 that use six samples of the sine function as coefficients areplotted in FIG. 21 . These interpolation weights approximate thefrequency response e^(jεω) with minimum least-square error across theentire frequency range [−π, +π] (in units of radians per unittime-step). The distortion plots in FIG. 21 show the ratio of theapproximate response to the ideal response. As |ω|→π the distortionincreases because the approximate responses are periodic but the idealresponse is not periodic. The distortion at lower frequencies isexaggerated because coefficients that minimize least-square error acrossthe entire frequency band tend to overemphasize the importance of thelarge error at high frequencies.

Much better results can be obtained by choosing the coefficients tominimize the least-square error in a reduced range of frequencies. Theupper left panel of FIG. 22 shows the distortion that results when sixcoefficients are chosen to minimize the least-square error in the band[−0.75π, +0.75π]. The distortion of these coefficients is less than 3%in the reduced band. Figures other three panels in FIG. 22 show thefurther reduction in distortion that results when 8, 10, or 12coefficients are used instead of 6. These results indicate that skewtransformations that are implemented using interpolations with 10 ormore coefficients will generate very accurate rotations if the initialdata is band-limited to three-quarters of the Nyquist frequency. Theoperation count for a two-dimensional rotation using 10 coefficientinterpolations is 60 floating-point operations per point (Eq. (7)), andthe operation count for a three-dimensional rotation using 10coefficient interpolations is 100 floating-point operations per point.This operation count is comparable to the cost (per point) of a singlethree-dimensional FFT, and is also comparable to the cost ofthree-dimensional rotations that interpolate using arrays of 4×4×4=64coefficients (e.g., bi-cubic interpolation). These interpolation schemeshave a computational cost of 128 floating-point operations per point andhave substantially more distortion than the 10-point skewtransformations.

GPU Implementation of Skew Transformations

The execution time of a skew transformation implemented on a GPU isprobably not determined by the operation count. Two other factors thatmust also be considered are:

-   -   1. The time required to upload data to the GPU    -   2. The time required to access data in GPU memory

Both of these operations can run in parallel with the arithmeticoperations, and both operations probably require more time than thearithmetic operations. Data transfers to and from the GPU are the mosttime consuming operations, but the extra time needed for these transferscan be used to perform other parallel computations such as split-steppropagations. However, the time spent loading data from GPU memory intoregisters for a skew transformation can only be parallelized witharithmetic operations for that transformation. Thus, the kernels used toimplement the skew transformations should be designed to minimize thenumber of times that each data value is loaded from memory. A kerneldesign that only requires each input data value to be loaded once andthat also only requires each output value to be stored once isillustrated below.

$\begin{matrix} ( {Input}arrow{{reg}\; 0} )\Rightarrow\begin{pmatrix}{{reg}\; 1} & {= {{Coeff}\; 1 \times {reg}\; 0}} \\{{{reg}\; 2} +} & {= {{Coeff}\; 2 \times {reg}\; 0}} \\{{{reg}\; 3} +} & {= {{Coeff}\; 3 \times {reg}\; 0}} \\{{{reg}\; 4} +} & {= {{Coeff}\; 4 \times {reg}\; 0}} \\{{{reg}\; 5} +} & {= {{Coeff}\; 5 \times {reg}\; 0}} \\{{{reg}\; 6} +} & {= {{Coeff}\; 6 \times {reg}\; 0}} \\{{{reg}\; 7} +} & {= {{Coeff}\; 7 \times {reg}\; 0}} \\{{{reg}\; 8} +} & {= {{Coeff}\; 8 \times {reg}\; 0}} \\{{{reg}\; 9} +} & {= {{Coeff}\; 9 \times {reg}\; 0}} \\{{reg}\; 0} & {= {{Coeff}\; 0 \times {reg}\; 0}}\end{pmatrix}\Rightarrow{\begin{pmatrix} {{reg}\; 9}arrow{Ouput}  \\ {{reg}\; 8}arrow{{reg}\; 9}  \\ {{reg}\; 7}arrow{{reg}\; 8}  \\ {{reg}\; 6}arrow{{reg}\; 7}  \\ {{reg}\; 5}arrow{{reg}\; 6}  \\ {{reg}\; 4}arrow{{reg}\; 5}  \\ {{reg}\; 3}arrow{{reg}\; 4}  \\ {{reg}\; 2}arrow{{reg}\; 3}  \\ {{reg}\; 1}arrow{{reg}\; 2}  \\ {{reg}\; 0}arrow{{reg}\; 1} \end{pmatrix}.}  & (13)\end{matrix}$The function of this kernel is to perform a fractional translation of aone-dimensional array using the 10 coefficient interpolation filterdescribed in the previous section. The registers are initially 0, andthe operation described by Eq. (13) is repeated for input values thatcyclically traverse the array in either the forward or backwarddirection, starting at a fixed index k₀. The indices of the outputvalues are always a fixed cyclical offset from the indices of the inputvalues. The direction of index progression and the initial index k₀ arechosen (based on the size and sign of the translation) to insure thatoutput values do not overwrite valid input values before the inputvalues have been processed. This kernel design is expected to reduce thecomputational time of each skew transformation to the time required tocopy the array inside the GPU.Dimensions of Rotated Arrays

The tissue parameters must be sampled in a rectangular volume thatcontains the entire hemisphere and must, therefore, have spatialdimensions of 15 cm×15 cm×7.5 cm. However, the initial estimates of thetissue parameters are not expected to have resolution better than 0.5mm. The dimensions of the tissue parameter arrays can, therefore, belimited to 300×300×150=13, 500,000 samples. This only requires about 54megabytes of memory per parameter if the parameter values are singleprecision floating-point numbers. To implement split-step computationsthese arrays must be upsampled to arrays of higher density. However,upsampling can take place after the tissue parameters have been rotatedto appropriate orientations. Thus, rotation of the tissue model is not acostly computation. Furthermore, each rotated and upsampled tissue modelis used for 256 forward computations.

However, rotation of the propagated fields back to a common grid iscostly because the field arrays are sampled more finely and everyforward solution will require rotation. Some reduction in the cost ofthese rotations can be realized by only rotating forward field valuesthat belong to the upper hemisphere. The blue dotted rectangles in FIG.23 show these reductions for propagations from different facetinclinations. The trapezoidal regions in the figure indicate limits ofthe directivity patterns of elements within the f acet. These ranges canalso be used to limit the rotated volumes, but these limits are do nottranslate into changes in array dimensions and would have to be builtinto the logic of the skew transformations.

These notes describe a method for accurate and efficient interpolationof a uniformly sampled time series x[n], n=0, ±1, ±2, . . . atfractional indices.

Any form of interpolation requires the time sequence to be composed ofsamples of a continuously varying signal with values at fractionalindices that can be inferred from the sampled values. The signal isusually assumed to be continuous and band limited to frequencies ω withmagnitudes that are less than the Nyquist frequency π. In that case, thecontinuous signal x(t) can be reproduced from the sequence x[n] by

$\begin{matrix}{{{x(t)} = {\sum\limits_{n = {- \infty}}^{\infty}{{x\lbrack n\rbrack}\frac{\sin( {\pi( {t - n} )} )}{\pi( {t - n} )}}}}.} & (1)\end{matrix}$Unfortunately, Eq. (1) is not an efficient way to evaluate x(t) becausethe sine function dies off rather slowly so that a large number of termsare needed to form accurate estimates.

The right side of Eq. (1) is a discrete inner product of the sequencex[n] and the sequence s_(t) [n]=sin(π(t−n))/π(t−n) that may also beevaluated in the frequency domain. Let X(ω): −π≤ω≤π denote the discretetime Fourier transform of x[n] and let S_(t) (ω): −π≤ω≤π denote thediscrete time Fourier transform of s_(t) [n] given by

$\begin{matrix}\begin{matrix}{{S_{t}(\omega)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{\frac{\sin( {\pi( {t - n} )} )}{\pi( {t - n} )}e^{{- j}\;\omega\; n}}}} \\{= {\sum\limits_{n = {- \infty}}^{+ \infty}{\frac{1}{2\pi}{\int\limits_{- \pi}^{+ \pi}{e^{{- j}\;{\omega{({t - n})}}}d\;\omega}}}}} \\{= {e^{{- j}\;\omega\; t}\frac{1}{2\pi}{\int\limits_{- \pi}^{+ \pi}{( {\sum\limits_{n = {- \infty}}^{+ \infty}e^{j\;\omega\; n}} )d\;\omega}}}} \\{= {{e^{{- j}\;\omega\; t}\frac{1}{2\pi}{\int\limits_{- \pi}^{+ \pi}{2\pi{\delta(\omega)}d\omega}}} = e^{{- j}\;\omega\; t}}}\end{matrix} & (2)\end{matrix}$Then, by Parseval's formula,

$\begin{matrix}{{x(t)} = {{\frac{1}{2\;\pi}{\int\limits_{- \pi}^{+ \pi}{{X(\omega)}{S_{t}(\omega)}d\;\omega}}} = {\frac{1}{2\;\pi}{\int\limits_{- \pi}^{+ \pi}{{X(\omega)}e^{{- j}\;\omega\; t}d\;{\omega.}}}}}} & (3)\end{matrix}$Equations (1) and (3) are time-domain and frequency-domain expressionsof harmonic interpolation that, as noted above, is exact when the samplesequence is not aliased. Using the inverse FFT algorithm to evaluate Eq.(3) is an efficient way to interpolate values at many locations that areall offset by the same amount from the nearest sample. However, Eq. (3)is not an efficient alternative to Eq. (1) if the spacing of theinterpolated samples is irregular.

Efficient interpolation schemes generally estimate x(t) as a linearcombination of a few of the sample values that are nearest to t. Forexample, if 0≤ε≤0.5, then a four-point interpolation of x(ε) has theformx(ε)≈w ⁻¹(ε)x[−1]+w ₀(ε)x[0]+w ₁(ε)x[1]+w ₂(ε)x[2].  (4)Applying the same weights to x[n−1], x[n], x[n+1], x[n+2] gives anestimate for x(n+ε) and applying the weights in reverse order gives anestimate for x(n+1−ε).

Equation (4) is also a discrete inner product that can be expressed inthe frequency domain as

$\begin{matrix}{{x(ɛ)} \approx {\frac{1}{2\;\pi}{\int\limits_{- \pi}^{+ \pi}{{{X(\omega)}\lbrack {{{w_{- 1}(ɛ)}e^{j\;\omega}} + {w_{0}(ɛ)} + {{w_{1}(ɛ)}e^{{- j}\;\omega}} + {{w_{2}(ɛ)}e^{{- 2}j\;\omega}}} \rbrack}d\;{\omega.}}}}} & (5)\end{matrix}$Equation (5) would give the same result as Eq. (3) (i.e., exact harmonicinterpolation) ifw ⁻¹(ε)e ^(jω) +w ₀(ε)+w ₁(ε)e ^(−jω) +w ₂(ε)e ^(−2jω) ≡e ^(−jωe).  (6)However, equality for all bi is not attainable in Eq. (6) when ε≠0, so aleast-square-error fit is appropriate. Since the exponential functionse^(jω), 1, e^(−jω), e^(−2jω) are orthogonal on the interval [−π, +π],the coefficients of this fit are given by

$\begin{matrix}{{w_{k}(ɛ)} = {{\frac{1}{2\pi}{\int\limits_{- \pi}^{+ \pi}{e^{j\;{\omega{({ɛ - k})}}}d\;\omega}}} = {\frac{\sin( {\pi( {ɛ - k} )} )}{\pi( {ɛ - k} )}.}}} & (7)\end{matrix}$This results in an imperfect interpolation that is characterized by thefrequency response

$\begin{matrix}{\frac{{{w_{- 1}(ɛ)}e^{j\;\omega}} + {w_{0}(ɛ)} + {{w_{1}(ɛ)}e^{{- j}\;\omega}} + {{w_{2}(ɛ)}e^{{- 2}\; j\;\omega}}}{e^{{- j}\;\omega\; ɛ}}.} & (8)\end{matrix}$The real and imaginary parts of t his distortion are shown in FIG. 24for ε=0.0, 0.1, 0.2, 0.3, 0.4, and 0.5. There is no distortion at allwhen ε=0.0, but the distortion increases as ε increases and is largestwhen ε=0.5. This distortion is caused by the discontinuity in theimaginary part of e^(−jωε) as ω wraps around from −π to +π. The value ofthe numerator in Eq. (8) is always the same at −π as the value at +πand, consequently, the fitted approximation to Eq. (6) becomes lessaccurate as the discontinuity increases (i.e., as ε→0.5).

The effect of the discontinuity may be reduced by fitting theinterpolation coefficients to W(ω)e^(−jωε) rather than e^(−jωε), whereW(ω) is a symmetric window function that tapers off as ω→±π. Four-pointinterpolations can reproduce the frequency response W(ω) e^(−jωε) withless distortion, and the response of the window W(ω) can be removed(without introducing any additional distortion) by frequency-domainprefiltering of x[n] by the inverse response 1/W (ω).

Selection of the function W (ω) is limited here by the requirement thatthe ε=0 interpolation response coincide with W (ω). This insures thatinterpolation of the prefiltered sample points will reproduce the samplevalues. Since W (ω) is a symmetric function the ε=0.0, filter weightsmust assign w₂ (0)=0 and w₁ (0)=w⁻¹ (0). This implies that the windowfunction W (ω) must have the form W (ω)=α+β cos ω. Furthermore, sincemultiplying W (ω) by a constant scale factor does not alter theinterpolation, the value of W (0) can be assumed to be 1, and thisimplies that β=α−1. Thus, the choice of W (ω) is reduced to theone-parameter family of functionsW _(α)(ω)=α+(1−α)cos ω.  (9)For each ε, the coefficients w⁻¹(ε), w₀(ε), w₁(ε), w₂(ε) are selected tominimize the square error in the approximationw ⁻¹(ε)e ^(jω) +w ₀(ε)+w ₁(ε)e ^(−jω) +w ₂(ε)e ^(−jω) ≈W _(α)(ω)e^(−jωε).  (10)This minimization problem has solution

$\begin{matrix}\begin{matrix}{{w_{k}(ɛ)} = {\frac{1}{2\pi}{\int\limits_{- \pi}^{+ \pi}{\lbrack {\alpha + {( {1 - \alpha} )\cos\;\omega}} \rbrack e^{{- j}{\omega{({ɛ - k})}}}d\;\omega}}}} \\{= {{\alpha\lbrack {\frac{1}{2\;\pi}{\int\limits_{- \pi}^{+ \pi}{e^{{- j}\;{\omega{({ɛ - k})}}}d\;\omega}}} \rbrack} + {\quad{\frac{1 - \alpha}{2}\lbrack {{\frac{1}{2\pi}{\int\limits_{- \pi}^{+ \pi}{e^{{- j}\;{\omega{({ɛ - k - 1})}}}d\omega}}} + {\frac{1}{2\;\pi}{\int\limits_{- \pi}^{+ \pi}{e^{{- j}\;{\omega{({ɛ - k + 1})}}}{d\omega}}}}}\; \rbrack}}}} \\{= {{\alpha\lbrack \frac{\sin( {\pi( {ɛ - k} )} )}{\pi( {ɛ - k} )} \rbrack} + {{\frac{1 - \alpha}{2}\lbrack {\frac{\sin( {\pi( {ɛ - k - 1} )} )}{\pi( {ɛ - k - 1} )} + \frac{\sin( {\pi( {ɛ - k + 1} )} )}{\pi( {ɛ - k + 1} )}} \rbrack}.}}}\end{matrix} & (11)\end{matrix}$The frequency response of the distortion for this interpolation schemeis

$\begin{matrix}{\frac{{{w_{- 1}(ɛ)}e^{j\;\omega}} + {w_{0}(ɛ)} + {{w_{1}(ɛ)}e^{{- j}\;\omega}} + {{w_{2}(ɛ)}e^{{- 2}j\;\omega}}}{{W_{\alpha}(\omega)}e^{{- j}\;\omega\; ɛ}}.} & (12)\end{matrix}$The real and imaginary parts of t his distortion are shown in FIG. 25for α=0.552 that was chosen (by visual inspection) to minimizedistortion. The distortion in FIG. 25 is substantially less than thedistortion in FIG. 24 . Even the curve for the ε=0.5 distortion deviatesfrom 1.0 by less than 4% for frequencies between −0.82 π and +0.82 π.

Some further improvement can be realized by reducing the emphasis of thefit a t the extreme edges of the frequency band. This may beaccomplished by only minimizing the square-error in Eq. (10) over thefrequency range [−λπ, +λπ] where λ<1. The computations are morecomplicated because the exponentials e^(jω), 1, e^(−jω), e^(−2jω) arenot orthogonal on this range. The solution is expressed in matrix formas

$\begin{matrix}{{\begin{pmatrix}{w_{- 1}(ɛ)} \\{w_{0}(ɛ)} \\{w_{1}(ɛ)} \\{w_{2}(ɛ)}\end{pmatrix} = {\begin{pmatrix}\langle {e^{j\;\omega},e^{j\;\omega}} \rangle & \langle {1,e^{j\;\omega}} \rangle & \langle {e^{{- j}\;\omega},e^{j\;\omega}} \rangle & \langle {e^{{- j}\; 2\;\omega},e^{j\;\omega}} \rangle \\\langle {e^{j\;\omega},1} \rangle & \langle {1,1} \rangle & \langle {e^{{- j}\;\omega},1} \rangle & \langle {e^{{- j}\mspace{11mu} 2\omega},1} \rangle \\\langle {e^{j\;\omega},e^{{- j}\;\omega}} \rangle & \langle {1,e^{{- j}\;\omega}} \rangle & \langle {e^{{- j}\;\omega},e^{{- j}\;\omega}} \rangle & \langle {e^{{- j}\; 2\;\omega},e^{{- j}\;\omega}} \rangle \\\langle {e^{j\;\omega},e^{{- j}\; 2\;\omega}} \rangle & \langle {1,e^{{- j}\; 2\;\omega}} \rangle & \langle {e^{{- j}\;\omega},e^{{- j}\; 2\;\omega}} \rangle & \langle {e^{{- j}\; 2\;\omega},e^{{- j}\; 2\;\omega}} \rangle\end{pmatrix}^{- 1} \times \begin{pmatrix}\langle {{{W_{\alpha}(\omega)}e^{{- j}\; ɛ\;\omega}},e^{j\;\omega}} \rangle \\\langle {{{W_{\alpha}(\omega)}e^{{- j}\; ɛ\;\omega}},1} \rangle \\\langle {{{W_{\alpha}(\omega)}e^{{- j}\; ɛ\;\omega}},e^{{- j}\;\omega}} \rangle \\\langle {{{W_{\alpha}(\omega)}e^{{- j}\; ɛ\;\omega}},e^{{- 2}\; j\;\omega}} \rangle\end{pmatrix}}},} & (13) \\{where} & \; \\{\langle {e^{{- j}l\;\omega},\ e^{{- j}k\;\omega}} \rangle = {{\frac{1}{2\;{\pi\lambda}}{\int\limits_{{- \lambda}\pi}^{{+ \lambda}\pi}{e^{{- {j{({l - k})}}}\omega}d\;\omega}}} = \frac{\sin( {\lambda{\pi( {l - k} )}} )}{\lambda{\pi( {l - k} )}}}} & (14) \\{and} & \; \\{\langle {{{W_{\alpha}(\omega)}e^{{- j}\; ɛ\;\omega}},e^{{- j}k\omega}} \rangle = {{\frac{1}{2\pi\lambda}{\int\limits_{{- \lambda}\pi}^{{+ \lambda}\pi}{\lbrack {\alpha + {( {1 - \alpha} )\cos\omega}} \rbrack e^{{- j}{\omega{({ɛ - k})}}}d\;\omega}}} = {{{\alpha\ \lbrack {\frac{1}{2\;\pi}{\int\limits_{{- \lambda}\pi}^{{+ \lambda}\pi}{e^{{- j}\;{\omega{({ɛ - k})}}}{d\omega}}}} \rbrack} + {\frac{1 - \alpha}{2}\lbrack {{\frac{1}{2\pi}{\int\limits_{{- \lambda}\pi}^{{+ \lambda}\pi}{e^{{- j}\;{\omega{({ɛ - k - 1})}}}d\;\omega}}} + {\frac{1}{2\pi}{\int\limits_{{- \lambda}\pi}^{{+ \lambda}\pi}{e^{{- j}\;{\omega{({ɛ - k + 1})}}}d\;\omega}}}} \rbrack}} = {\lambda\{ \ {{\alpha\ \lbrack \frac{\sin( {{\lambda\pi}( {ɛ - k} )} )}{{\lambda\pi}( {ɛ - k} )} \rbrack} + {\frac{1 - \alpha}{2}\lbrack {\frac{\sin( {{\lambda\pi}( {ɛ - k - 1} )} )}{{\lambda\pi}( {ɛ - k - 1} )} + \frac{\sin( {{\lambda\pi}( {ɛ - k + 1} )} )}{{\lambda\pi}( {ɛ - k + 1} )}} \rbrack}} \}}}}} & (15)\end{matrix}$and the real and imaginary parts of the distortion that result from theweights obtained for λ=0.8, α=0.56 (also chosen by visual inspection tominimize distortion) are shown in FIG. 26 .

This distortion should be acceptable for interpolations used inscattering calculations. However, if greater accuracy is required, thenharmonic doubling of the sampling rate may be applied prior to theapplication of the prefiltered interpolation technique described above.In that case, the frequency response of the interpolated samples will belimited to the range −π/2≤ω≤π/2 where the distortion in FIG. 26 isessentially negligible.

It will be appreciated that variants of the above-disclosed and otherfeatures and functions, or alternatives thereof, may be combined intomany other different systems or applications. Various presentlyunforeseen or unanticipated alternatives, modifications, variations, orimprovements therein may be subsequently made by those skilled in theart which are also intended to be encompassed by the following claims.

What is claimed is:
 1. A device for volumetric ultrasound imagingcomprising: a transducer array, substantially configured in the shape ofa faceted hemisphere to form a volumetric imaging region within a cavityof the faceted hemisphere, including transducer elements that transmitultrasound signals to and receive ultrasound signals from a scatteringobject within the faceted hemisphere; and a plurality ofdata-acquisition assemblies, each connected to the transducer array,configured to simultaneously transmit the received ultrasound signals inparallel to a network of processors, the network of processorscomprising: a plurality of acquisition and computation nodes, eachcomprising one or more graphical processing units (GPUs) that receivethe received ultrasound signals in parallel and process the receivedultrasound signals in parallel to obtain pulse-echo responses andreconstruct a volumetric image of the scattering object within theimaging region; and a control node that independently assigns delays toeach transducer element and coordinates firing sequences and samplingintervals of the transducer elements to co-register the pulse-echoresponses.
 2. The device of claim 1, wherein the transducer elements arearranged across the surface of a plurality of triangular planar facets.3. The device of claim 2, wherein some of the triangular planar facetsare equilateral triangles and some of the triangular planar facets areisosceles triangles.
 4. The device of claim 3, wherein: the isoscelestriangles are arranged to form pentagonal transducer element assemblies;and the pentagonal transducer element assemblies and the equilateraltriangles are arranged to form the faceted hemisphere.
 5. The device ofclaim 2, wherein the transducer elements are arranged pseudorandomlyacross the surfaces of the triangular planar facets.
 6. The device ofclaim 1, wherein the transducer elements are piezoelectric elements. 7.The device of claim 1, wherein at least one of the transducer elementsfurther comprises a diverging lens.
 8. The device of claim 1, wherein:the transducer array is configured to simultaneously transmit encodedtransmissions from sets of the transducer elements; and the pulse echoresponses are obtained by decoding the encoded transmissions.
 9. Thedevice of claim 1, further comprising: hardware cables thatindependently transfer the received ultrasound signals from eachdata-acquisition assembly to each processor in parallel.
 10. The deviceof claim 9, wherein each of the hardware cables connects one of theprocessors to two of the data-acquisition assemblies.
 11. A method forreconstructing a volumetric ultrasound image, the method comprising:providing an array of transducer subarrays that are substantiallyconfigured in the shape of a faceted hemisphere to form a volumetricimaging region within a cavity of a faceted hemisphere; transmittingultrasound signals to and receiving ultrasound signals from a scatteringobject within the imaging region by transducer elements of thetransducer subarrays; receiving the ultrasound signals from thetransducer subarrays, in parallel, by a plurality of data-acquisitionassemblies connected in parallel to the transducer subarrays;transmitting the ultrasound signals, in parallel, by the plurality ofdata-acquisition assemblies to a plurality of acquisition andcomputation nodes of a network of processors, each acquisition andcomputation node including one or more graphical processing units(GPUs); processing the received ultrasound signals in parallel, by theacquisition and computation nodes, to obtain pulse-echo responses andperform parallel construction of a volumetric image of the scatteringobject within the imaging region the pulse-echo responses; andcoordinating firing sequences and sampling intervals of the transducerelements, by a control node of the network of processors, to co-registerthe pulse-echo responses.
 12. The method of claim 11, wherein thetransducer elements are arranged across the surface of a plurality oftriangular planar facets.
 13. The method of claim 12, wherein some ofthe triangular planar facets are equilateral triangles and some of thetriangular planar facets are isosceles triangles.
 14. The method ofclaim 13, wherein: the isosceles triangles are arranged to formpentagonal transducer element assemblies; and the pentagonal transducerelement assemblies and the equilateral triangles are arranged to formthe faceted hemisphere.
 15. The method of claim 12, wherein thetransducer elements are arranged pseudorandomly across the surfaces ofthe triangular planar facets.
 16. The method of claim 11, wherein thetransducer elements are piezoelectric elements.
 17. The method of claim11, wherein at least one of the transducer elements further comprises adiverging lens.
 18. The method of claim 11, wherein transmitting theultrasound signals and obtaining the pulse-echo responses comprises:simultaneously transmitting encoded transmissions from sets of thetransducer elements; and decoding the encoded transmissions.
 19. Themethod of claim 11, wherein the received ultrasound signals areindependently transferred from each data-acquisition assembly to eachprocessor in parallel via hardware cables.
 20. The method of claim 19,wherein each of the hardware cables connects one of the processors totwo of the data-acquisition assemblies.